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Chapter

6 Rules for Exponents and theReasons for Them

6.1 INTEGER POWERS AND THE EXPONENT RULESRepeated addition can be expressed as a product. For example,

Similarly, repeated multiplication can be expressed as a power. For example,

Here, 2 is called the base and 5 is called the exponent. Notice that 25 is not the same as 52, because25 = 2 × 2 × 2 × 2 × 2 = 32 but 52 = 5 × 5 = 25.

In general, if a is any number and n is a positive integer, then we define

Notice that a1 = a, because here we have only 1 factor of a. For example, 51 = 5. We call a2 the square of a and a3 thecube of a.

Multiplying and Dividing Powers with the Same Base

When we multiply powers with the same base, we can add the exponents to get a more compact form. For example,52 · 53 = (5 · 5) · (5 · 5 · 5) = 52 + 3 = 55. In general,

Thus,

Example 1 Write with a single exponent:

(a) q5 · q7

(b) 62 · 63

(c) 2n · 2m

(d) 3n · 34

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(e) (x + y)2(x + y)3.

SolutionUsing the rule an · am = an + m we have

(a) q5 · q7 = q5 + 7 = q12

(b) 62 · 63 = 62 + 3 = 65

(c) 2n · 2m = 2n + m

(d) 3n · 34 = 3n + 4

(e) (x + y)2(x + y)3 = (x + y)2 + 3 = (x + y)5.

Just as we applied the distributive law from left to right as well as from right to left, we can use the rulean · am = an + m written from right to left as an + m = an · am.

Example 2 Write as a product:

(a) 52 + a

(b) xr + 4

(c) yt + c

(d) (z + 2)z + 2.

SolutionUsing the rule an + m = an · am we have

(a) 52 + a = 52 · 5a = 25 · 5a

(b) xr + 4 = xr · x4

(c) yt + c = yt · yc

(d) (z + 2)z + 2 = (z + 2)z · (z + 2)2.

When we divide powers with a common base, we subtract the exponents. For example, when we divide 56 by 52, weget

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More generally, if n > m,

Thus,

Example 3 Write with a single exponent:

(a)

(b)

(c) , where n > 4

(d)

(e) .

SolutionSince we have

(a)

(b)

(c)

(d)

(e) .

Just as with the products, we can write in reverse as .

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Example 4 Write as a quotient:

(a) 102 - k

(b) eb - 4

(c) zw - s

(d) (p + q)a - b.

SolutionSince we have

(a)

(b)

(c)

(d)

Raising a Power to a Power

When we take a number written in exponential form and raise it to a power, we multiply the exponents. For example,

More generally,

Thus,

Example 5 Write with a single exponent:

(a) (q7)5

(b) (7p)3

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(c) (ya)b

(d) (2x)x

(e)

(f) .

SolutionUsing the rule (am)n = am · n we have

(a) (q7)5 = q7 · 5 = q35

(b) (7p)3 = 73p

(c) (ya)b = yab

(d)

(e)

(f) .

Example 6 Write as a power raised to a power:

(a) 23 · 2

(b) 43x

(c) e4t

(d) .

SolutionUsing the rule am · n = (am)n we have

(a) 23 · 2 = (23)2. This could also have been written as (22)3.(b) 43x = (43)x, which simplifies to 64x. This could also have been written as (4x)3.(c) e4t = (e4)t. This could also have been written as (et)4.(d)

Products and Quotients Raised to the Same Exponent

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When we multiply 52 · 42 we can change the order of the factors and rewrite it as52 · 42 = (5 · 5) · (4 · 4) = 5 · 5 · 4 · 4 = (5 · 4) · (5 · 4) = (5 · 4)2 = 202. Sometimes, we want to use this process inreverse: 102 = (2 · 5)2 = 22 · 52.

In general,

Thus,

Example 7 Write without parentheses:

(a) (qp)7

(b) (3x)n

(c) (4ab2)3

(d) (2x2n)3n.

SolutionUsing the rule (ab)n = anbn we have

(a) (qp)7 = q7p7

(b) (3x)n = 3nxn

(c) (4ab2)3 = 43a3(b2)3 = 64a3b6

(d) .

Example 8 Write with a single exponent:

(a) c4d4

(b) 2n · 3n.(c) 4x2

(d) a4(b + c)4

(e) (x2 + y2)5(c - d)5.

SolutionUsing the rule anbn = (ab)n we have

(a) c4d4 = (cd)4

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(b) 2n · 3n = (2 · 3)n = 6n

(c) 4x2 = 22x2 = (2x)2

(d) a4(b + c)4 = (a(b + c))4

(e) .

Division of two powers with the same exponent works the same way as multiplication. For example,

Or, reversing the process,

More generally,

Thus,

Example 9 Write without parentheses:

(a)

(b)

(c)

(d) .

SolutionUsing the rule we have

(a)

(b)

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(c)

(d) .

Example 10 Write with a single exponent:

(a)

(b)

(c)

(d)

(e)

SolutionUsing the rule we have

(a)

(b)

(c)

(d)

(e) .

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Zero and Negative Integer Exponents

We have seen that 45 means 4 multiplied by itself 5 times, but what is meant by 40, 4-1 or 4-2? We choose definitionsfor exponents like 0, -1, -2 that are consistent with the exponent rules.

If a ≠ 0, the exponent rule for division says

But , so we define a0 = 1 if a ≠ 0. The same idea tells us how to define negative powers. If a ≠ 0, the exponent

rule for division says

But , so we define a-1 = 1/a. In general, we define

Note that a negative exponent tells us to take the reciprocal of the base and change the sign of the exponent, not tomake the number negative.

Example 11 Evaluate:

(a) 50

(b) 3-2

(c) 2-1

(d) (-2)-3

(e)

Solution (a) Any nonzero number to the zero power is one, so 50 = 1.(b) We have

(c) We have

(d) We have

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(e) We have

With these definitions, we have the exponent rule for division, where n and m are integers.

Example 12 Rewrite with only positive exponents. Assume all variables are positive.

(a)

(b)

(c)

(d)

Solution (a) We have

(b) We have

(c) We have

(d) We have

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In part (a) of Example 12, we saw that the x-2 in the denominator ended up as x2 in the numerator. In general:

Example 13 Write each of the following expressions with only positive exponents. Assume all variablesare positive.

(a)

(b)

(c)

(d)

(e)

Solution (a) .

(b) .

(c) .

(d) .

(e) .

Summary of Exponent Rules

We summarize the results of this section as follows.

general

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Expressions with a Common Base

If m and n are integers,

1. an · am = an + m

2.

3. (am)n = am · n

Expressions with a Common Exponent

If n is an integer,

1. (ab)n = anbn

2.

Zero and Negative Exponents

If a is any nonzero number and n is an integer, then:

• a0 = 1•

Common Mistakes

Be aware of the following notations that are sometimes confused:

For example, -24 = -(24) = -16, but (-2)4 = (-2)(-2)(-2)(-2) = 16.

Example 14 Evaluate the following expressions for x = -2 and y = 3:

(a) (xy)4

(b) -xy2

(c) (x + y)2

(d) xy

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(e) -4x3

(f) -y2.

Solution (a) (-2 · 3)4 = (-6)4 = (-6)(-6)(-6)(-6) = 1296.(b) -(-2) · (3)2 = 2 · 9 = 18.(c) (-2 + 3)2 = (1)2 = 1.(d) (-2)3 = (-2)(-2)(-2) = -8.(e) -4(-2)3 = -4(-2)(-2)(-2) = 32.(f) -(3)2 = -9.

Problems for Section 6.1

EXERCISES

Evaluate the expressions in Exercises 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 without using a calculator.

1. 3 · 23

Answer:

24

2. -32

3. (-2)3

Answer:

-8

4. 51 · 14 · 32

5. 52 · 22

Answer:

100

6.

7. (-5)3 · (-2)2

Answer:

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-500

8. -53 · -22

9. -14 · (-3)2(-23)

Answer:

72

10.

11. 30

Answer:

1

12. 03

In Exercises 13, 14, 15, 16, 17, 18, 19, 20, 21 and 22, evaluate the following expressions for x = 2, y = -3, andz = -5.

13. -xyz

Answer:

-30

14. yx

15. -yx

Answer:

-9

16.

17.

Answer:

-8/125

18. x-z

19. -x-z

Answer:

-32

20.

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21.

Answer:

729/1000

22.

In Exercises 23, 24, 25, 26, 27, 28, 29, 30 and 31 , write the expression in the form xn, assuming x ≠ 0.

23. x3 · x5

Answer:

x8

24.

25. (x4 · x)2

Answer:

x10

26.

27.

Answer:

x2

28.

29. (x3)5

Answer:

x15

30.

31.

Answer:

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x5

In Exercises 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44 and 45, write with a single exponent.

32. 42 · 4n

33. 2n22

Answer:

2n + 2

34. a5b5

35.

Answer:

(a/b)x

36.

37.

Answer:

22n - m

38. An + 3BnB3

39. BaBa + 1

Answer:

B2a + 1

40. (x2 + y)3(x + y2)3

41.

Answer:

(x + y)20

42. 162y8

43.

Answer:

(g + h)3

44.

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45.

Answer:

(a + b)3

Without a calculator, decide whether the quantities in Exercises 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58and 59 are positive or negative

46. (-4)3

47. -43

Answer:

Negative

48. (-3)4

49. -34

Answer:

Negative

50. (-23)42

51. -3166

Answer:

Negative

52. 17-1

53. (-5)-2

Answer:

Positive

54. -5-2

55. (-4)-3

Answer:

Negative

56. (-73)0

57. -480

Answer:

Negative

58. (-47)-15

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59. (-61)-42

Answer:

Positive

In Exercises 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71 and 72, write each expression without parentheses.Assume all variables are positive.

60.

61.

Answer:

c12/d4

62.

63.

Answer:

64r6/125s12

64.

65.

Answer:

36g10/49h14

66. (cf)9

67. (2p)5

Answer:

32p5

68.

69.

Answer:

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16tb4t

70.

71.

Answer:

3 · 16x e4x

72.

PROBLEMS

In Problems 73, 74, 75, 76 and 77, decide which expressions are equivalent. Assume all variables are positive.

73. (a) 3-2

(b)

(c)

(d)

(e)

Answer:

(a), (c), (d) equivalent; (b), (e) equivalent

74. (a)

(b)

(c)

(d)

(e)

75. (a)

(b)

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(c)

(d) x-r

(e)

Answer:

(a), (c), (e) equivalent; (b), (d) equivalent

76. (a)

(b)

(c)

(d)

(e)

77. (a)

(b)

(c)

(d)

(e)

Answer:

(a), (b) equivalent; (c), (d), (e) equivalent

In Problems 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90 and 91 , write each expression as a product or aquotient. Assume all variables are positive.

78. 32 + 3

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79. a4 + 1

Answer:

a4 + 1 = a4 · a

80. e2 + r

81. 104 - z

Answer:

104/10z

82. ka - b

83. 4p + 3

Answer:

4p · 43

84. 6a - 1

85. (-n)a + b

Answer:

(-n)a(-n)b

86. xa + b + 1

87. p1 - (a + b)

Answer:

p/(papb)

88. (r - s)t + z

89. (p + q)a - b

Answer:

(p + q)a/(p + q)b

90. et - 1(t + 1)

91. (x + 1)ab + c

Answer:

(x + 1)ab(x + 1)c

In Problems 92, 93, 94, 95, 95, 96, 97 and 98, write each expression as a power raised to a power. There may bemore than one correct answer.

92. 42 · 4

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93. 23x

Answer:

(23)x = 8x

94. 52y

95. 34a

Answer:

96.

97. 3e2t

Answer:

98. (x + 3)2w

99. If 3a = w, express 33a in terms of w.

Answer:

w3

100. If 3x = y, express 3x + 2 in terms of y.

101. If 4b = c, express 4b - 3 in terms of c.

Answer:

c/64

102. If , and z = xc, what is a?

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