Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 1

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 2

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 3

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 4

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 5

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 6

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 7

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 8

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 9

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 10

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 11

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 12

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 13

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 14

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 15

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 16

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 17

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 18

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 19

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 20

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 21

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 22

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 23

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 24

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 25

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 26

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 27

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 28

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 29

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 30

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 31

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 32

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 33

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 34

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

eSolutions Manual - Powered by Cognero Page 35

7-3 Rational Exponents

Write each expression in radical form, or write each radical in exponential form.

1. 

SOLUTION:  

2. 

SOLUTION:  

3. 

SOLUTION:  

4. 

SOLUTION:  

Simplify.

5. 

SOLUTION:  

6.  

 

SOLUTION:  

7. 

SOLUTION:  

8. 

SOLUTION:  

9. 

SOLUTION:  

10. 

SOLUTION:  

11. 

SOLUTION:  

12. 

SOLUTION:  

Solve each equation.

13. 

SOLUTION:  

  Therefore, the solution is 4.

14. 

SOLUTION:  

  Therefore, the solution is 1.

15. 

SOLUTION:  

  Therefore, the solution is 5.5.

16. CCSS TOOLS  A weir is used to measure water flow in a channel. (Refer to the photo on page 410)   For a rectangular broad crested weir, the flow Q in cubic feet per second is related to the weir length L in feet and

height H of the water by . Find the water height for a weir that is 3 feet long and has flow of 38.4 cubic feet per second.

SOLUTION:  

  Therefore, the water height for the weir is 4 feet. 

Write each expression in radical form, or write each radical in exponential form.

17. 

SOLUTION:  

18. 

SOLUTION:  

19. 

SOLUTION:  

20. 

SOLUTION:  

21. 

SOLUTION:  

22. 

SOLUTION:  

23. 

SOLUTION:  

24. 

SOLUTION:  

Simplify.

25. 

SOLUTION:  

26. 

SOLUTION:  

27. 

SOLUTION:  

28. 

SOLUTION:  

29. 

SOLUTION:  

30. 

SOLUTION:  

31. 

SOLUTION:  

32. 

SOLUTION:  

33. 

SOLUTION:  

34. 

SOLUTION:  

35. 

SOLUTION:  

36. 

SOLUTION:  

37. 

SOLUTION:  

38. 

SOLUTION:  

39. 

SOLUTION:  

40. 

SOLUTION:  

41. 

SOLUTION:  

42. 

SOLUTION:  

43. 

SOLUTION:  

44. 

SOLUTION:  

Solve each equation.

45. 

SOLUTION:  

  Therefore, the solution is 5.

46. 

SOLUTION:  

  Therefore, the solution is 2.

47. 

SOLUTION:  

  Therefore, the solution is .

48. 

SOLUTION:  

  Therefore, the solution is .

49. 

SOLUTION:  

  Therefore, the solution is .

50. 

SOLUTION:  

  Therefore, the solution is .

51. 

SOLUTION:  

  Therefore, the solution is 8.

52. 

SOLUTION:  

  Therefore, the solution is 2.

53. 

SOLUTION:  

  Therefore, the solution is 8.

54. 

SOLUTION:  

  Therefore, the solution is .

55. 

SOLUTION:  

  Therefore, the solution is .

56. 

SOLUTION:  

  Therefore, the solution is .

57. CONSERVATION Water collected in a rain barrel can be used to water plants and reduce city water use. Water

flowing from an open rain barrel has velocity , where v is in feet per second and h is the height of the water in feet. Find the height of the water if it is flowing at 16 feet per second.

SOLUTION:  

  The height of the water is 4 feet.

58. ELECTRICITY The radius r in millimeters of a platinum wire L centimeters long with resistance 0.1 ohm is

. How long is a wire with radius 0.236 millimeter?

SOLUTION:  

  So, the wire has a length of 16 centimeters.

Write each expression in radical form, or write each radical in exponential form.

59. 

SOLUTION:  

60. 

SOLUTION:  

61. 

SOLUTION:  

62. 

SOLUTION:  

63. 

SOLUTION:  

64. 

SOLUTION:  

65. 

SOLUTION:  

66. 

SOLUTION:  

Simplify.

67. 

SOLUTION:  

68. 

SOLUTION:  

69. 

SOLUTION:  

70. 

SOLUTION:  

71. 

SOLUTION:  

72. 

SOLUTION:  

73. 

SOLUTION:  

74. 

SOLUTION:  

75. 

SOLUTION:  

76. 

SOLUTION:  

77. 

SOLUTION:  

78. 

SOLUTION:  

Solve each equation.

79. 

SOLUTION:  

  Therefore, the solution is 12.

80. 

SOLUTION:  

  Therefore, the solution is 3.

81. 

SOLUTION:  

  Therefore, the solution is –5.

82. 

SOLUTION:  

  Therefore, the solution is .

83. 

SOLUTION:  

  Therefore, the solution is .

84. 

SOLUTION:  

  Therefore, the solution is 11.

85. CCSS MODELING  The frequency f in hertz of the nth key on a piano is .

a. What is the frequency of Concert A?  b. Which note has a frequency of 220 Hz?

SOLUTION:  a. Replace n with 49 in the equation to determine the frequency of Concert A.

  So, the frequency of Concert A is 440 hertz.   b. Replace f with 220 in the equation to determine the value of n.

  So, the note with a frequency of 220 corresponds to the 37th note on the piano or the A below middle C.

86. RANDOM WALKS Suppose you go on a walk where you choose the direction of each step at random. The path of a molecule in a liquid or a gas, the path of a foraging animal, and a fluctuating stock price are all modeled as random walks. The number of possible random walks w of n steps where you choose one of d directions at each

step is .

a. How many steps have been taken in a 2-direction random walk if there are 4096 possible walks?  b. How many steps have been taken in a 4-direction random walk if there are 65,536 possible walks?  c. If a walk of 7 steps has 2187 possible walks, how many directions could be taken at each step? 

SOLUTION:  a.  Replace w with 4096 and d with 2 in the equation to determine the value of n.

So, in a 2-direction random walk having 4096 possible walks there have been 12 steps taken.  b. Replace w with 65,536 and d with 4 in the equation to determine the value of n.

  So, in a 4-direction random walk having 65,536 possible walks there have been 8 steps taken.  c. Replace n with 7 and w with 2187 in the equation to determine the value of d.

  So, in 2187 possible walks of 7 steps there could be 3 directions taken at each step.

87. SOCCER The radius r of a ball that holds V cubic units of air is modeled by .  

  What are the possible volumes of each size soccer ball?

SOLUTION:  Size 3: d = 7.3 in.. r = 3.65

  Size 3: d= 7.6 in., r = 3.8

  Size 4: d = 8.0 in.. r = 4.0

  Size 4: d= 8.3 in., r = 4.15

  Size 5: d = 8.6 in.. r = 4.3

  Size 5: d= 9.0 in., r = 4.5

  Volume of size 3 soccer balls vary from 204.0 to 230.2 cubic inches. Volume of size 4 soccer balls vary from 268.5to 299.9 cubic inches. Volume of size 5 soccer balls vary from 333.6 to 382.4 cubic inches.

88. MULTIPLE REPRESENTATIONS In this problem, you will explore the graph of an exponential function. a. TABULAR Copy and complete the table below.  

  b. GRAPHICAL Graph f (x) by plotting the points and connecting them with a smooth curve. c. VERBAL Describe the shape of the graph of f (x). What are its key features? Is it linear?

SOLUTION:  a.

  b. 

c. The graph of f (x) = 4

x is a curve. It has no x-intercept, a y-intercept of 1, the domain is all real numbers, the

range is all positive real numbers, it is increasing over the entire domain, as x approaches infinity f (x) approaches infinity, as x approaches negative infinity f (x) approaches 0. The graph is not linear.

89. OPEN ENDED Write two different expressions with rational exponents equal to .

SOLUTION:  Sample answer:   First:

  Second:

 

Thus can be represented by  and  .

90. CCSS ARGUMENTS Determine whether each statement is always, sometimes, or never true.Assume that x is a nonnegative real number. Explain your reasoning.  

a.

b.

c.

d.

e.

f.

SOLUTION:  

a. Sometimes  is true. It is only true when x = 1.

b. Sometimes  is true.  It is only true when x = 1.

c. Sometimes  is true.  It is only true when x = 1.

d. is always true by definition of .

e.  is always true since   =  = x1 or x.

f. Sometimes  is true.  It is only true when x = 1.

91. CHALLENGE For what values of x is ?

SOLUTION:  

  Thus, the expressions will be equal for x = –1, 0, 1.

92. ERROR ANALYSIS Anna and Jamal are solving . Is either of them correct? Explain your reasoning.  

SOLUTION:  

  Anna is correct. Jamal did not write the expressions with equal bases before applying the Power Property of Equality.

93. WRITING IN MATH Explain why 2 is the principal fourth root of 16.

SOLUTION:  

Sample answer: 2 is the principal fourth root of 16 because 2 is positive and 24 = 16.

94. What is the value of ?

  A 5 B 11 C 25 D 35

SOLUTION:  

  Therefore, the correct choice is D.

95. At a movie theater, the costs for various numbers of popcorn and hot dogs are shown.

Which pair of equations can be used to find p , the cost of a box of popcorn, and h, the cost of a hot dog?F

    G

    H

    J

 

SOLUTION:  Let p be the cost of a box of popcorn and h the cost of a hot dog. Then p + h = 8.50 and 2p + 4h = 21.6.The correct choice is G.  

96. SHORT RESPONSE Find the dimensions of the rectangle if its perimeter is 52 inches. 

SOLUTION:  

  Therefore, the dimensions of the rectangle are 17.5 inches by 8.5 inches.

97. If 34 = 9

x, then x = 

  A 1 B 2 C 4 D 5

SOLUTION:  

  So, the correct choice is B.

Simplify each expression. Assume that no denominator equals zero. 

98. 

SOLUTION:  

99. 

SOLUTION:  

100. 

SOLUTION:  

101. 

SOLUTION:  

102. 

SOLUTION:  

103. 

SOLUTION:  

104. GARDENING Felipe is planting a flower garden that is shaped like a trapezoid as shown. 3

Use the formula to find the area of the garden.

 

SOLUTION:  Replace h with 3a, b1 with 6a, and b2 with 4a in the formula to determine the value of A.

 

 

Therefore, the area of the garden is 15a2 square units.

Write each equation in slope-intercept form.

105. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 3x – 1.

106. 

SOLUTION:  

  So, in slope-intercept form the equation is y = 6x + 11. 

107. 

SOLUTION:  

  So, in slope-intercept form the equation is y = –2x – 12.

108. 

SOLUTION:  

  So, in slope-intercept form the equation is .

109. 

SOLUTION:  

  So, in slope-intercept form the equation is .

110. 

SOLUTION:  

  So, in slope-intercept form the equation is .

Find each power.

111. 103

SOLUTION:  

112. 105

SOLUTION:  

113. 10–1

SOLUTION:  

114. 

SOLUTION:  

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7-3 Rational Exponents