Name:__________________________________________ ALSO PRINT YOUR NAME AT TOP OF COVER PAGE OF UNIT. Videos produced on YouTube by YourMathGal Julie Harland that accompany sections in this Unit are at https://sites.google.com/site/packet2int/home/p2videos To navigate to that site, go to YourMathGal.com. Click Home. Click My Books on side bar. Click Math 64. Click Unit 1. Click Videos for Unit 2. Before doing the problems in each section, watch and transcribe the videos accompanying that section for your notes. *An asterisk next to a problem indicates its solution is worked out on a video. Intermediate Algebra Unit 2 Radicals, Rational Exponents, Complex Numbers 2.1 Square Roots .................................................................................................... 2 2.2 Higher Order Roots .................................................................................... 12 2.3 Add and Multiply Radicals ....................................................................... 16 2.4 Divide Radicals ............................................................................................. 27 2.5 Rational Exponents .................................................................................... 41 2.6 Radical Equations ........................................................................................ 54 2.7 Pythagorean Theorem & Distance Formula .................................... 64 2.8 Complex Numbers ...................................................................................... 72 2.9 Dividing Complex Numbers .................................................................... 85 2.10 Review Exercises ......................................................................................... 93 ISBN: 978-‐0-‐8401-‐3255-‐0 Printing Date: February 2016 This Unit is (c) 2015 Julie Harland and is licensed under a Creative Commons Attribution-‐NonCommercial-‐ShareAlike 4.0 International License
Page 2 of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra Intermediate Algebra
Print First and Last Name:_____________________________________________________________ 2.1 Square Roots 1. Evaluate the following:
*a. 02 = *b. 32 = *c. −2( )2 =
*d. 56
⎛⎝⎜
⎞⎠⎟2
= *e. 30( )2 = f. −40( )2 =
*g. .4( )2 = h. .07( )2 = *i. −3( )2 =
j. −5( )2 = k. 52 = l. 202 = 2. State the two square roots of each number. *a. 16: _____________ b. 100: _____________ 3. What symbol is used to denote the “principal” square root of a number and what
does “principal square root of a number” mean? 4. Simplify. If it’s not a real number, write NOT REAL. Each problem has one answer.
*a. 16 = *b. 81 = *c. 36 =
d. 0 = e. 1 = *f. −9 =
*g. − 49 = h.
100121
= *i.
916
=
j. 2500 = k. − 64 = l. −49 =
Page 3 of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra Intermediate Algebra
Square Roots with variables Simplify . Assume all variables are positive.
*5. x2( )2 = *6.
x3( )2 =
*7. x8( )2 = 8. x5 ⋅ x5 =
*9. x4 = *10. n16 =
*11. x6 = *12. x16 =
13. m8 = 14. m12 =
*15. b30 = 16. y22 =
*17. 16x16 = *18. 36m
10 =
*19. 9m8n4y2 =
21. 25m24n6 =
23. 81a100b60c40 =
24. Give 3 examples of irrational numbers involving square roots.
Page 4 of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra Intermediate Algebra
Approximating Square Roots 25. List the perfect squares from 22 up to 122 . Start with 4, 9, etc. ________________________________________________________________________________________________ Use a calculator to approximate each square root to the 3 places after the decimal point. Then square the approximation and round to 5 places after the decimal point.
*26. 15 ≈ ________________ Square that approximation:_____________________________
*27. 30 ≈ ________________ Square that approximation: _____________________________
28. 41 ≈ ________________ Square that approximation: _____________________________
*29. Enter 3 in a calculator, round to 3 places after the decimal point and write on the first blank below. Then multiply by 2. Then enter 2 3 in a calculator and round.
a. 2 3( ) ≈ 2 · ________________ = __________________ b. 2 3 ≈ ____________________
30. Enter 5 in a calculator, round to 3 places after the decimal point and write on the first blank below. Then multiply by 3. Then enter 3 5 in a calculator and round.
a. 3 5( ) ≈ 3 · ________________ = __________________ b. 3 5 ≈ ____________________
Page 5 of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra Intermediate Algebra
Multiplication Property of Square Roots *31. Complete the following statements.
a. If x ≥ 0, then x2 = _______ b. x x = ___ c. x( )2 = ___ 32. What is the main difference between part a above compared to part b and c? 33. Evaluate the following. All of these are in the form shown in b or c above.
*a. 7 7 = *b. 103 103 =
c. 65 65 = *d. junk junk =
*e. 5x3y 5x3y = f. 2x − 5 2x − 5 =
*g. stuff( )2 =
*h. 19( )2 =
i. m3 + 5n( )2 = j. 101( )2 =
k. name( )2 =
l. math math = Even though the square root of a negative number is not a real number, you can still compute the following using part b and c from question 1 to simplify these. Your answer should be a negative number. We’ll cover square roots of negative numbers later.
m. −7( )2 =
n. −14 −14 =
Page 6 of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra Intermediate Algebra
MULTIPLICATION PROPERTY FOR SQUARE ROOTS
If a ≥ 0 and b ≥ 0 , then a ⋅ b = ab To simplify things and make directions in this unit less cumbersome, we will make the assumption that all variables in this unit are positive unless otherwise stated, and write the
multiplication property as a ⋅ b = ab Multiply. Then simplify if possible. Assume variables are positive.
*34. 3 ⋅ 7 = *35. 2 ⋅ 8 =
*36. 5 ⋅ 5 = *37. 47 ⋅ 47 =
*38. 3 ⋅ 5 = 39. 11 ⋅ 2 =
*40. a ⋅ a5 = 41. x3 ⋅ x5 =
*42. 2 ⋅ 3 ⋅ 7 = 43. 5 ⋅ 3 ⋅ 2 =
*44. 2x3 ⋅ 18x5 = 45. 2m
7 ⋅ 8m9 =
*46. 23x5 ⋅ 23x7 =
47. 17ab3 ⋅ 17a7b =
48. 27x5y15 ⋅ 3x9y =
49. x3 2x5 ⋅3 8x9 =
Page 7 of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra Intermediate Algebra
Simplifying Square Roots A square root of a counting number is considered simplified if no factors greater than 1 are perfect squares. The number under the square root symbol is called the radicand. If the radicand has a perfect square factor, the square root of that number can be simplified. 50. List all factors for each radicand, and circle any perfect squares greater than 1. Then state if the given square root is simplifed
*a. 15 : State the factors of 15: ______________________________________________________
Is 15 simplified? ___________
*b. 12 : State the factors of 12: ______________________________________________________
Is 12 simplified? ___________
c. 30 : State the factors of 30: ______________________________________________________
Is 30 simplified? ___________
d. 45 : State the factors of 45: ______________________________________________________
Is 45 simplified? ___________
*51. Show all steps to simplify 12
Page 8 of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra Intermediate Algebra
Show steps to simplify. Use a calculator to verify the approximation of the original square root and the simplified answer are the same. Put a BOX around each answer.
*52. 18
53. 75
*54. 50
55. 40
*56. 54
*57. 72
58. 28
59. 99
*60. 3 75
61. 5 24
62. 2 27
Page 9 of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra Intermediate Algebra
Show steps to simplify. . Assume variables are positive Put a BOX around each answer.
*63. x15
*64. m11
65. n33
66. a2m7
67. x13y6
68. 44m11
*69. 48x8n5
70. 98m10n3
71. 22a9n11
Page 10 of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra
*72. −6 45x3
*73. 4x 28x2
74. 3m 32m5
*75. 4x2m 63x2m7
*76. −5x4 8x2y9
77. x3n 18x8n15
78. −5x3n 20x5n7
Page 11 of 98 Unit 2 Harland 2.1 Square Roots Intermediate Algebra
Multiply and simplify. Show Steps. Assume variables are positive. Put BOX around answer.
*79. 21 ⋅ 14
*80. 35 ⋅ 55
81. 30 ⋅ 42
82. 30 ⋅ 33
83. 22x3 33x
*84. 6x3 ⋅ 2x5 ⋅ 5x7
*85. 12x3 ⋅ 3x
*86. −3x2 10 ⋅5 30
87. −5x2 6x4 ⋅2x3 10x5
Page 12 of 98 Unit 2 Harland 2.2 Higher Order Roots Intermediate Algebra
2.2 Higher Order Roots
If a ≥ 0 , then ann = a Simplify. If it’s not a real number, write “not a real number”. Assume variables are positive. Show all steps starting with #16. Pay close attention to the index, which is the little number you see on the left (except on the square root, where it is optional to write the index of 2). So for #1, the index is 3, which means you are taking the cube root of 64.
*1. 643 2. 814
*3. 325
4. −273 *5. 100004
9. −16
*10. m8( )3 *11. m243 12. x183
13. n505 14. n147
15. n124
*16. 403
*17. 543
18. 804
19. 2503
*20. m143
21. x135
Page 13 of 98 Unit 2 Harland 2.2 Higher Order Roots Intermediate Algebra
22. n314
*23. 563
*24. x103
*25. 24x103
*26. x263
27. x10m125
28. −3x 4x143
*29. 80x12y154
30. 5xy −40x20y193
Page 14 of 98 Unit 2 Harland 2.2 Higher Order Roots Intermediate Algebra
We have been using this definition: If a ≥ 0 , then ann = a But what if we are not sure whether a is positive or negative? It depends on the index, n. Below is the definition which is true regardless of the sign of a.
If n is an even integer, then ann = a If n is an odd integer, then ann = a Simplify. Use absolute value signs as needed. Do not assume variables are positive. Do not leave negative numbers under the radical sign. Write “NOT REAL if it is not a real number. BOX answer.
*31. b33 *32. m44
33. n99 34. m66
*35. 164 *36. −83
37. −3( )44 38. −19
*39. −16 40*. −814
41. −1 42. 766
*43. −1253 *44. −23
45. m20n84 46. m20n305
Page 15 of 98 Unit 2 Harland 2.2 Higher Order Roots Intermediate Algebra
Simplify. Use absolute value signs as needed. Do not assume variables are positive. Do not leave negative numbers under the radical sign. BOX answer.
*47. −250x6y103
48. −32x11y155
*49. 45x3y9
50. −63x26y93
51. 48x10y154
52. 64x12y155
Page 16 of 98 Unit 2 Harland 2.3 Add and Multiply Radicals Intermediate Algebra
2.3 Add and Multiply Radicals Simplify if possible. If already simplified, write “SIMPLIFIED”. Assume variables are positive. Show all steps starting with #13. BOX answer.
*1. 3c + 5c *2. 3 2 + 5 2
*3. 3 7 + 5 7 4. 6x + x
5. 6 5 + 5 *6. 2 3 + 5 2
7. 6 5m − 3 5m 8. b −10b
9. 2 −10 2 10. 2 2x − 3 2x
11. 7 3y − 2 3y + 3y
12. 7 3n − 2 n + 3n
*13. 8 + 18
14. 12 + 27
15. 2 27 + 3 50
16. 4 75 − 50
Page 17 of 98 Unit 2 Harland 2.3 Add and Multiply Radicals Intermediate Algebra
*17. 5x 45x3 − 3 20x5
18. 5 8x + 2 18x − 50x
*19. 3m 18x2 − x 27m2
20. 3 12x2 − 2x 27
*21. 4x 32 − 18x2 + 2x 128
Page 18 of 98 Unit 2 Harland 2.3 Add and Multiply Radicals Intermediate Algebra
22. 8x 75m + 3 50m
23. 7 20x5 − x 24x3
24. 3m 45x8 − x3 54x2m2
25. 3m 28 − 20 + 44m2
Page 19 of 98 Unit 2 Harland 2.3 Add and Multiply Radicals Intermediate Algebra
Adding Higher Order Roots Simplify if possible. Assume variables are positive. Show all steps. BOX answer.
26. 8 73 + 4 73
27. 135 − 4 135
28. 24 + 5 23 − 3 2
29. 9 163 + 3 23
*30. 5 543 + 3 2503
31. 3 323 + 5 1083
*32. 18 + 503
33. 98 + 163
Page 20 of 98 Unit 2 Harland 2.3 Add and Multiply Radicals Intermediate Algebra
*34. 10 403 − 5 6253
35. 5 563 − 70003
*36. 8 x94 − x2 81x4
37. 8 x135 − x 16x74
*38.
995x
− 44x2
39.
455x2
− 80x4
Page 21 of 98 Unit 2 Harland 2.3 Add and Multiply Radicals Intermediate Algebra
Multiplying Binomials with Radicals Use the distributive Property to Multiply and Simplify. Show all steps. BOX answer.
40. 2 2 + 3( )
41. 5 4 + 5( )
42. 7 2 7 + 3( )
43. 5 8 2 − 3 5( )
44. 2 7 3 − 7( )
Page 22 of 98 Unit 2 Harland 2.3 Add and Multiply Radicals Intermediate Algebra
*45. 2 3 11 + 3 3( )
46. 3 5 2 + 5( )
47. 4 5 2 + 3 5( )
*48. 2 6 5 2 − 3 18( )
49. 3 10 5 2 − 15( )
Page 23 of 98 Unit 2 Harland 2.3 Add and Multiply Radicals Intermediate Algebra
Multiply and Simplify. You can use the FOIL method or any method for multiplying binomials.
50. 5 − 7( ) 3+ 7( )
51. 4 + 3 5( ) 2 + 5( )
*52. 2 + 3 5( ) 3− 4 5( )
53. 1− 5( ) 2 + 5( )
54. 2 2 − 5( ) 2 + 3 5( )
Page 24 of 98 Unit 2 Harland 2.3 Add and Multiply Radicals Intermediate Algebra
55. 5 2 + 6( ) 2 + 2 6( )
56. 2 − 3 5( ) 2 + 3 5( )
57. 4 7 + 3 2( ) 4 7 − 3 2( )
*58. 2 14 3 21 − 8 35( )
59. 5 6 42 − 66( )
Page 25 of 98 Unit 2 Harland 2.3 Add and Multiply Radicals Intermediate Algebra
*60. 5 + 2 7( )2
61. 5 3 − 2( )2
62. 5 3 − 2( ) 5 3 + 2( )
*63. 2 3 − 5( ) 3 + 3 5( )
*64. 3− 5( ) 3+ 5( )
Page 26 of 98 Unit 2 Harland 2.3 Add and Multiply Radicals Intermediate Algebra
65. 5 − 11( ) 5 + 11( )
66. 5 − 11( )2
67. 2 5 + 7( ) 2 5 − 7( )
68. 2 5 + 7( )2
Page 27 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
2.4 Divide Radicals
Quotient Rule for Square Roots: ab= a
b
Use the Quotient Rule to simplify the following. The answer should not contain any square roots in the denominator. All square roots should be as simplified as possible, and all fractions should be reduced. Show all steps.
*1.
169
*2.
1825
3.
1336
4. 3 1736
5.
40x5
10x
6.
40x7
5x
*7.
12x3y5
3xy
Page 28 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
Rationalize the denominator, and simplify. Use the quotient rule as needed. The answer should not contain any square roots in the denominator. All square roots should be as simplified as possible, and all fractions should be reduced. Show all steps. BOX answer.
*8.
12
9.
13
*11.
35
*12.
126
13. − 1015
14.
3 56
Page 29 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
15.
53
16.
45
17.
67
*18.
512
*19.
78
*20.
245
Page 30 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
21.
5x2
18
22.
350x
23.
23 5
24.
3 10m3
5m 6
25.
7 2015
Page 31 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
Dividing Higher Order Roots
If an and bn are real numbers, and bn ≠ 0 , then ab
n = an
bn.
Simplify. Show all steps. Rationalize the denominator and/or use the quotient rule as needed. The answer should not contain any radicals in the denominator. All radicals should be as simplified as possible, and all fractions should be reduced. Show steps. BOX answer.
*26.
827
3
27.
64125
3
*28. − 827
3
*29.
14
3
30.
249
3
31.
25
3
Page 32 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
*32.
363
33.
5103
*34.
32
4
35.
23
4
*36.
59
3
37.
349
3
Page 33 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
38.
3m10
85
*39.
13y10
3
40.
1ab2c34
Page 34 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
2.15 Conjugates & Rationalizing Denominators Simplify. Show steps. Put a BOX around answer.
*41. 5 + 7( ) 5 − 7( )
42. 4 − 7( ) 4 + 7( )
43. 7 5 − 7( )
44. 5 7( ) 7
45. 4 − 7( ) 7
46. 4 7( ) 7
Page 35 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
Simplify. Show steps. Put a BOX around answer.
*47. 3 2 − 4( ) 3 2 + 4( )
*48. 7 − 2( ) 7 + 2( )
Fill in the blanks.
49. The conjugate of 2 + 7 is ______________________
50. The conjugate of 2 3 − 5 is ______________________
51. The conjugate of 2 3 + 5 2 is ______________________ 52. The conjugate of a + b is ______________________ Rationalize the denominator and simplify. Show all steps. There should be no square roots in the denominator. Reduce if possible. BOX answer.
*53. 1
3− 2
*54. 13 2
Page 36 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
55. 43 −1
56. 43
57. 9
2 + 7
58. 92 7
Page 37 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
59. 1
2 3 − 5
*60. 5
11 − 2 2
61. 6
2 3 + 5 2
Page 38 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
*62. 8
3 2 − 2
*63. 3+ 24 − 5
64. 3+ 24 5
*65. 7 + 37 − 3
Page 39 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
66. 3 23− 6
67. 10
10 − 6
68. 10 + 2 21
4
*67. 5 2 + 32
Page 40 of 98 Unit 2 Harland 2.4 Divide Radicals Intermediate Algebra
Below are some more problems, including some more challenging problems to do on your own paper. The challenging problems are especially appropriate for students planning to enroll in courses such as College Algebra, Business Calculus, or Precalculus.
70. 4 − 3 23 2
71. 3 2
4 + 3 2
72. 2 + 32 − 3
73. 12 − 3 2
−6
74. −1510 − 5
75. 5 6 − 320
76. 10 3 + 2 21
6
Page 41 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
2.5 Rational Exponents Simplify. Use Laws of Exponents to simplify.
*1. b0 = *2. b−2 = Use the Laws of Exponents to complete the right side.
*3. bm ⋅bn = 4. bm( )n = Simplify. Show the intermediate step using one of the appropriate Laws from above.
*5. 512 ⋅5
12 = 6. 3
12 ⋅3
12 =
*7.
512
⎛⎝⎜
⎞⎠⎟
2
8.
1013
⎛⎝⎜
⎞⎠⎟
3
9. 1013 ⋅10
13 ⋅10
13 =
Simplify.
*10. 5 ⋅ 5 = 11. 73( )3 = 12. Fill in the blank to complete the definition below.
If n is a positive integer greater than 1 and bn is a real number, then b1n = ________
Page 42 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
For each expression, state the base, and then circle the correct way to write it using radical notation. Then simplify the correct one.
13. −121( )12 Base: ________ Circle and simplify correct one below.
−121 − 121 12⋅121
*14. −100013 Base: ________ Circle and simplify correct one below.
−10003 − 10003
*15. 144− 12 Base: ________ Circle and simplify correct one below.
− 12⋅144 − 144 −144 −1
144
1144
16. −216( )−13 Base: ________ Circle and simplify correct one below.
1−2163
− 2163 −2163 − 13⋅216 1
2163
17. −8− 13 Base: ________ Circle and simplify correct one below.
− 83 1−83
− 183 −83
Page 43 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
Write each expression using radical notation, and simplify if possible. If there are negative exponents, rewrite the problem using positive exponents before writing using radical notation. If the answer is not a real number, state “NOT REAL”. Show all steps. BOX answer.
*18. 1612
*19. 1013
20. 8114
21. −25( )12
*22. 8112
23. 6413 =
24. 4912
*25. −2713
26. −34313
*27. 16− 12
Page 44 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
28. 9− 12
29. 8− 13
30. −1000− 13
31. −125( )−13
32. m12
*33. −9( )12
34. 11−13
35. n15 =
Page 45 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
36. −2512
37. −27( )13
*38. 5x14
39. −7x13
40. 7x−13
Page 46 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
Simplifying with Rational Exponents There are two ways to rewrite an expression with a rational exponent, where the denominator n is not zero. The second way is usually easier to compute with numbers.
*41. Follow the rule above to show two ways to rewrite bmn without using fractional
exponents—use a radical and integer exponent. Assume m and n are positive integers greater than 1 and that m/n is reduced.
bmn =
*42. Use both ways above to rewrite the expression. Then use each way to simplify 823 .
a. 823 =
b. 823
*43. Use both ways above to rewrite the expression. Then use each way to simplify 932 .
a. 932 =
b. 932 =
44. When simplifying 932 without a calculator, did you find it easier to take the square
root of 9 first and then cube it, or to cube 9 and then take the square root? Explain.
Page 47 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
Below is a Summary of the Laws of Exponents. Assume a and b (the bases) are positive.
Note bm
bn can be written two ways. They are equivalent. So bm−n = 1
bn−m.
Often we write it the first way shown if m>n and the second way shown if n>m so that the exponent will be positive.
Also note the two ways to write amn . They are equivalent. So an( )m = amn .
bm ⋅bn = bm+n
bm( )n = bmnbm
bn= bm−n = 1
bn−m
ab( )n = anbn
ab
⎛⎝⎜
⎞⎠⎟n
= an
bn
b−n = 1bn
b0 = 1
b1n = bn
amn = an( )m = amn
ab
⎛⎝⎜
⎞⎠⎟−n
= bn
an
Page 48 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
Write each expression using radical notation in two ways. Then, simplify one of the notations. Pick the one you find easiest to compute. Show all steps. BOX answer.
45. 2723
46. 2532
47. 8134
Write each expression using radical notation, and simplify if possible. If there are negative exponents, first rewrite the problem using positive exponents. Then rewrite using radical notation. If the answer is not a real number, state NOT REAL. Show all steps. BOX answer.
*48. −1634
49. −2532
50. −25( )32
Page 49 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
*51. 8− 23
52. 4− 52
*53. −27( )43
54. −32( )25
*55. 49
⎛⎝⎜
⎞⎠⎟
32
56. − 278
⎛⎝⎜
⎞⎠⎟
23
Page 50 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
*57. − 452
58. −952
*59. −36( )12
60. 4925
⎛⎝⎜
⎞⎠⎟− 32
61. − 278
⎛⎝⎜
⎞⎠⎟− 23
Page 51 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
*62. 2x( )35
63. 3x( )54
*64. 2x35
65. 3x54
*66. −27( )−23
67. − 8125
⎛⎝⎜
⎞⎠⎟− 23
*68. x− 16
69. m− 25
Page 52 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
*70. 4
5x− 12
*71. − 4− 32
72. 2x − 3( )23
Use the Laws of Exponents to write each expression with a base of x—simply one base of x raised to some power. After you do that, if the exponent is negative, rewrite with a positive exponent, and if the exponent is fractional, write in radical form. Show steps. Box answer.
*73. x34 ⋅ x
− 74
74. x415 ⋅ x
715
*75.
x34
x16
76.
m710
m110
Page 53 of 98 Unit 2 Harland 2.5 Rational Exponents Intermediate Algebra
Advanced Rational Exponents Below are some challenging problems you might try. These are appropriate for students planning to enroll in courses such as College Algebra, Business Calculus, or Precalculus. Assume all variables are positive. Use the laws of exponents to simplify each expression. Do not write any answers with negative exponents. Show all steps. BOX Answer.
*77. 27u3( )23 *78.
x14 x
− 12
x23
79. x16x
− 56
x13
*80. a−2b3( )
18
a−3b( )−14
81. mb m−1b3( )0
m−2b4( )−12
82. 8u3
125m9
⎛⎝⎜
⎞⎠⎟
−23
*83. a318 84. a525
*85. 364 86. x1015
*87. x + 4( )48 Rewrite using rational exponents. Then write so the base is raised to a single power. Then write in radical form. Assume all variables are positive. Show all steps. BOX answer.
*88. y23 ⋅ y6 *89. x5
x6
90. n23 nn4
*91. 23 5
*92. x34 y 23
Page 54 of 98 Unit 2 Harland 2.6 Radical Equations Intermediate Algebra
2.6 Radical Equations POWER RULE FOR EQUATIONS: If both sides of an equation are raised to the same power, all the solutions of the original equation are also among the solutions of the new equation. The Power Rule does NOT state that both equations necessarily have the same solution. Any time you raise both sides of an equation to the same power, the new equation might have more solutions than the original equation. This is an extremely important point, and why it’s especially crucial that if you use the power rule to solve an equation that you always check any solutions in the ORIGINAL equation. To solve an equation containing one or more radicals, we use the Power Rule for Equations. How to Solve an Equation containing one or more Radicals Step 1: Isolate one radical on one side of the equation. Step 2: Use the Power Rule to raise each side of the equation to the power of the index of
the radical—if it’s a square root, square both sides; if it’s a cube root, cube both sides, etc.
Step 3: Simplify both sides of the equations. Cautionary Note: If one side contains more than
one term, and you are squaring that side, you do not simply square each term. You might be squaring a binomial on that side, in which case, use the FOIL method.
Step 4: If after simplifying each side, the equation has another radical, repeat steps 1-‐3. Step 5: Once there are no more radicals, solve the equation. Step 6: Check each solution in the ORIGINAL equation. The last step is crucial and very important when using the Power Rule. Always Check! If a proposed solution does not check in the original equation, we call it an extraneous root. It is NOT a solution unless it checks in the original.
Page 55 of 98 Unit 2 Harland 2.6 Radical Equations Intermediate Algebra
Solve each equation, and check each solution. Write solutions that check on the blank space provided. Show all steps, including steps to checking each solution.
*1. x = 4 Solution(s): __________________ Show check(s).
2. 2 m = 14 Solution(s): __________________ Show check(s).
*3. x + 4 = 5 Solution(s): __________________ Show check(s).
4. 3− 2x = 2 Solution(s): __________________ Show check(s).
*5. x = −3 Solution(s): __________________ Show check(s).
Page 56 of 98 Unit 2 Harland 2.6 Radical Equations Intermediate Algebra
6. x + 9 = 2 Solution(s): __________________ Show check(s).
*7. x − 3 − 2 = 4 Solution(s): __________________ Show check(s).
8. x +1 − 3 = 2 Solution(s): __________________ Show check(s).
*9. 2x − 3 + 3 = 6 Solution(s): __________________ Show check(s).
Page 57 of 98 Unit 2 Harland 2.6 Radical Equations Intermediate Algebra
10. 2x +1 +1= 4 Solution(s): __________________ Show check(s).
*11. 5x − 9 = 2x − 3 Solution(s): __________________ Show check(s).
12. 2x − 73 = 8 − 3x3 Solution(s): __________________ Show check(s).
*13. x + 5 = x +1 Solution(s): __________________ Show check(s).
Page 58 of 98 Unit 2 Harland 2.6 Radical Equations Intermediate Algebra
14. x − 8 = x − 2 Solution(s): __________________ Show check(s).
*15. x + x + 5 = 7 Solution(s): __________________ Show check(s).
16. 2 x + 3 +1= x + 4 Solution(s): __________________ Show check(s).
Page 59 of 98 Unit 2 Harland 2.6 Radical Equations Intermediate Algebra
*17. 2x3 + 5 = 9 Solution(s): __________________ Show check(s).
18. 3 x3 = x2 +17x3 Solution(s): __________________ Show check(s).
*19. x − 43 − 5 = −7 Solution(s): __________________ Show check(s).
20. 2x + 33 + 3 = 5 Solution(s): __________________ Show check(s).
Page 60 of 98 Unit 2 Harland 2.6 Radical Equations Intermediate Algebra
*21. 7x − 5 = 5 − 7x Solution(s): __________________ Show check(s).
22. x + 84 = 2x4 Solution(s): __________________ Show check(s).
*23. x + 34 − 5 = −3 Solution(s): __________________ Show check(s).
24. 2x + 34 + 9 = 12 Solution(s): __________________ Show check(s).
Page 61 of 98 Unit 2 Harland 2.6 Radical Equations Intermediate Algebra
*25. x − x + 9 = 1 Solution(s): __________________ Show check(s).
26. x − 7 + x = 7 Solution(s): __________________ Show check(s).
*27. 2x −14 − x = −1 Solution(s): __________________ Show check(s).
Page 62 of 98 Unit 2 Harland 2.6 Radical Equations Intermediate Algebra
28. x − 8 = x − 2 Solution(s): __________________ Show check(s).
*29. x − 4 + x = 6 Solution(s): __________________ Show check(s).
*30. x + 2 + 4 = x Solution(s): __________________ Show check(s).
Page 63 of 98 Unit 2 Harland 2.6 Radical Equations Intermediate Algebra
31. 26 −11x = 4 − x Solution(s): __________________ Show check(s).
*32. −3x +16 +1= −4x + 25 Solution(s): __________________ Show check(s).
Page 64 of 98 Unit 2 Harland 2.7 Pythagorean Theorem & Distance Formula Intermediate Algebra
2.7 Pythagorean Theorem & Distance Formula The Pythagorean Theorem is a very famous mathematical theorem. In the diagram on the left below, look at the triangle in the middle. One leg has length a, and the other leg has length b, and the hypotenuse has length c. The square with side of a has an area of a2, the square with side of b has an area of b2, and the square with side of c has an area of c2. The Pythagorean Theorem states that the sum of the areas of the two smaller squares is the same area as the larger square. The Pythagorean Theorem is often stated this way: The square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (called legs). If we are looking at a triangle where the lengths of the legs are a and b, and the length of the hypotenuse is c, we often write this as shown on the diagram on the right.
There are several proofs showing why the Pythagorean Theorem is true. I’ve include one proof on a video. You can check find other proofs by searching the internet. 1. Draw a right triangle with legs of lengths x and y, and hypotenuse of length z. Then
use the Pythagorean Theorem to state the equation that is true for this triangle using x, y, and z.
Page 65 of 98 Unit 2 Harland 2.7 Pythagorean Theorem & Distance Formula Intermediate Algebra
For each right triangle having the sides indicated, draw a picture of the triangle, and use the Pythagorean Theorem to find the missing side. Write the exact answer (simplifying any radicals if possible), and then use a calculator to round the answer to one place after the decimal point. Show work. *2. One leg is 5” and hypotenuse is 8”. Exact length of other leg:_____________ Rounded length of other leg:_____________ 3. One leg is 7’ and hypotenuse is 15’. Exact length of other leg:_____________ Rounded length of other leg:_____________
Page 66 of 98 Unit 2 Harland 2.7 Pythagorean Theorem & Distance Formula Intermediate Algebra
*4. The legs are 3 m and 5 m Exact length of hypotenuse:_____________ Rounded length of hypotenuse:_____________ 5. The legs are 7 cm and 15 cm Exact length of hypotenuse:_____________ Rounded length of hypotenuse:_____________ *6. One leg is 8 cm; hypotenuse is 12 cm. Exact length of other leg:_____________ Rounded length of other leg:_____________
Page 67 of 98 Unit 2 Harland 2.7 Pythagorean Theorem & Distance Formula Intermediate Algebra
*7. The legs are 7 m and 24 m Exact length of hypotenuse:_____________ Rounded length of hypotenuse:_____________ *8. One leg is2 2 ; hypotenuse is 8 Exact length of other leg:_____________ Rounded length of other leg:_____________
9. The legs are 5 and 2 3 Exact length of hypotenuse:_____________ Rounded length of hypotenuse:_____________
Page 68 of 98 Unit 2 Harland 2.7 Pythagorean Theorem & Distance Formula Intermediate Algebra
DISTANCE FORMULA If (x1.y1) and (x2.y2) are two points, the distance, d, between the two points, is shown below. This can be shown to be true by using the Pythagorean Theorem. An picture of how to derive the Distance Formula is below.
*10. State the distance formula between points x1, y1( ) and x2, y2( ) . 11. State the distance formula between points a,b( ) and c,d( ) .
Page 69 of 98 Unit 2 Harland 2.7 Pythagorean Theorem & Distance Formula Intermediate Algebra
Plot the 2 points and draw a line segment between them and call it d. Then compute the exact distance, d, between the points, simplifying any radical if possible. Then use a calculator to round to the nearest tenth—one place after decimal point. Show all steps. *12. −3,5( ) and 6,−2( ) Exact: ___________ Rounded: ____________
*13. −3,−6( ) and 4,−8( ) Exact: ___________ Rounded: ____________
14. −1,3( ) and 2,4( ) Exact: ___________ Rounded: ____________
x
y
5
5
- 5
- 5
x
y
5
5
- 5
- 5
x
y
5
5
- 5
- 5
Page 70 of 98 Unit 2 Harland 2.7 Pythagorean Theorem & Distance Formula Intermediate Algebra
*15. −6,−5( ) and −8,−8( ) Exact: ___________ Rounded: ____________
*16. 0,4( ) and −9,3( ) Exact: ___________ Rounded: ____________
*17. 12,−3⎛
⎝⎜⎞⎠⎟ and 4
12,6⎛
⎝⎜⎞⎠⎟ Exact: ___________ Rounded: ____________
x
y
5
5
- 5
- 5
x
y
5
5
- 5
- 5
x
y
5
5
- 5
- 5
Page 71 of 98 Unit 2 Harland 2.7 Pythagorean Theorem & Distance Formula Intermediate Algebra
*18. −2,4( ) and −2,−6( ) Exact: ___________ Rounded: ____________
Use the distance formula to find the exact distance between each pair of points. Simplify any radical if possible. Then use a calculator to round to the nearest tenth—one place after decimal point. Show all steps. 19. 5,−5( ) and −7,6( ) Exact: ___________ Rounded: ____________ 20. −4,−9( ) and −2,−3( ) Exact: ___________ Rounded: ____________
21. 12,−1⎛
⎝⎜⎞⎠⎟ and − 1
2,7⎛
⎝⎜⎞⎠⎟ Exact: ___________ Rounded: ____________
x
y
5
5
- 5
- 5
Page 72 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
2.8 Complex Numbers Imaginary Number and Powers of i *1. Define the imaginary unit, i : i = _________ *2. Evaluate: i2 = ________ *3. Is i a real number?
*4. Is −9 a real number?
*5. Is −9 in its most simplified form?
*6. Show the steps to simplify: −9
*7. If −9 is not a real number, what kind of number is it?
*8. Complete the statement: If b > 0 , then −b = ___________ It is NEVER simplified to leave a negative number under a square root symbol. Anytime there is a negative number under a square root, the first step is to rewrite it as the square root of the absolute value of that number (so it’s positive) times the square root of -‐1—which is i . The i is NOT under the square root symbol. To clarify this, some people write the i in front of the square root if the square root does not simplify to a rational number. See below for an example:
−18 = 18 ⋅ −1 = 3 2i Note the i is NOT under the square root sign. Be careful NOT to write 3 2i Or you may do it this way:
−18 = −1 ⋅ 18 = i ⋅3 2 = 3i 2
So you may see −18 written either of these ways in simplified form: 3i 2 or 3 2i You need to get used to seeing it in either form. It is easier to pick out the imaginary part if the i is at the end, but make sure you do not extend the square root symbol over the i .
Page 73 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
Simplify. There should be no negative numbers under the square root, and all radicals need to be simplified. Show all steps.
*9. −25
*10. −18
11. −90
12. −28 Any number that can be written in the the form bi where b is a real number and i = −1 is called an imaginary number (or pure imaginary number). Any number that can be written in the form a + bi where a and b are real numbers and i = −1 is an imaginary number is called a complex number. a is called the real part and b is called the imaginary part of a + bi . Note that the imaginary part is only the real number b , not bi . 13. Below are examples of numbers that can be put in the form a + bi so it is easy to
identify the real and imaginary parts. See the examples at the beginning and then fill in the real and imaginary parts for the rest. Simplify a and b as needed when you put it in the form a + bi so that the real and imaginary parts are simplified.
Number In form a + bi Real part Imaginary Part 3− 5i 3− 5i 3 −5 2 3 2 3 + 0i 2 3 0 8i 0 + 8i 0 8 −7 + i −7 + i or −7 +1i −7 1 −3+ 13i −3+ 13i
13 − i
−2 5i
17
−36
5 − −16
75 + −13
Page 74 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
Simplify each of the following. Answer to each will either be 1, −1 , i , or −i . Show your reasoning and steps. 14. i2 *15. i3
*16. i4 *17. i5
*18. i6 *19. i7
20. i8 *21. i20
*22. i32 23. i40
*24. i22 *25. i33
*26. i103 27. i17
*28. i57 *29. i30
30. i102 *31. i23
*32. i44 33. i501
*34. i50 35. i69
These next two are a little tricky. You can do them!
*36. i−1 37. i−2
Page 75 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
Below are some important definitions and properties. It’s very important to note the property below is only true if both a and b are positive.
If a ≥ 0 and b ≥ 0 , then a ⋅ b = ab For any numbers a and b, positive or negative, the following are always true.
ab= a
b a a = a a( )2 = a Definition of i (Note that i is NOT a variable) : i = −1 and i2 = −1
If b > 0 , then −b = bi or −b = i b If there is a negative number under a square root, use the above property for the first step. If a negative number is under a square root, the FIRST STEP is to rewrite the expression using i so only a positive number is under the square root. Then, if possible, simplify the square root using properties of square roots.
Example: −6 = 6i NOTE: i is NOT under the square root symbol. You can also write it this way: −6 = i 6 Simplify. If a negative number is under a square root, the FIRST STEP is to rewrite the expression using i so only a positive number is under the square root. Then, if possible, simplify the square root using properties of square roots. Show steps. BOX answer.
*38. −50
39. −15
40. −90
*41. −6 −9
Page 76 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
Simplify. If a negative number is under a square root, the FIRST STEP is to rewrite the expression using i so only a positive number is under the square root. Then, if possible, simplify the square root using properties of square roots. Show steps. BOX answer.
*42. 5 −45
43. 2 −54
For any number, a a = a and a( )2 = a 44. In the following few examples, you may use the properties above. These are special
cases where you do not have to first rewrite it with i since the properties are true for all values of a. You will get the same answer using the properties above, or by first rewriting each square root with i and a positive number under the square root before simplifying.
*a. −11 ⋅ −11
*b. −41 ⋅ −41
c. −73 ⋅ −73
d. −26( )2 This property applies only when both a and b are positive: a ≥ 0 and b ≥ 0 , then a ⋅ b = ab Simplify. If a negative number is under a square root, the FIRST STEP is to rewrite the expression using i so only a positive number is under the square root. Then, if possible, simplify the square root using properties of square roots. Show steps. BOX answer.
*45. −2 ⋅ −8
46. −27 ⋅ −3
Page 77 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
47. −5 ⋅ −6
*48. 2 −12
49. −5 −63
*50. −6 ⋅ −10
51. −10 ⋅ −15
*52. 3 −7 ⋅2 −14
53. −5 −21 ⋅2 −15
Page 78 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
Product Rule: If a ≥ 0 and b ≥ 0 , then a ⋅ b = ab
Quotient Rule: ab= a
b
*54. −182
55. −18−2
*56. −40−5
57. −120−15
58. −5 33 ⋅4 −22
Page 79 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
Add complex numbers like you add like terms even though i is not a variable. Add the real numbers together, and add the imaginary numbers together. Simplify the real and
imaginary parts as needed.
To subtract, distribute the negative sign: a + bi( )− c + di( ) = (a − c)+ (b − d)i Add or Subtract as indicated. Simplify the real and imaginary parts. Write your answer as a real number, a pure imaginary number, or as a complex number in the form a + bi or a − bi . Show steps. BOX answer. *59. 3− 2i( ) + 6 + 4i( ) 60. −3+ 7i( ) + −4 − 7i( ) 61. 3+ 2i( ) + −3− 5i( ) 62. 3+ 2i( )− −3+ i( )
63. 6 3 + 2i( ) + −3 3 + 5 2i( )
64. 6 3 + 2i( )− 3 3 − 5 2i( )
65. 3− −9( )− 2 + −16( )
66. 5 3 − −8( ) + 12 + −2( )
a + bi( ) + c + di( ) = (a + c)+ (b + d)i
Page 80 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
Multiply. Simplify the real and imaginary parts. Do not leave i2 in answer as that can be simplified to -‐1. Write your answer as a real number, a pure imaginary number, or as a complex number in the form a + bi or a − bi . Show all steps. BOX answer. *67. −4i ⋅3i *68. −5 ⋅−2i 69. −i ⋅−i 70. −4i ⋅9
71. −9 ⋅2i 72. −7 9 − 7i( ) *73. 3i 5 + 7i( ) 74. −2i 4 + 3i( )
75. −6 4 + −6( ) *76. 2 + 3i( ) 6 − 5i( ) 77. −1− 2i( ) −1− i( )
Page 81 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
Remember: a + b( )2 = a + b( ) a + b( )
*78. 7 + i( )2 *79. 7 + i( ) 7 − i( )
*80. 2 + 3i( )2 81. 2 + 3i( ) 2 − 3i( )
82. 8 − i( )2
Page 82 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
*83. 4 + 3i( ) 4 − 3i( ) 84. −5 − 2i( ) −5 + 2i( ) *85. 3+ 5i( ) 3− 5i( )
86. 3− 5i( )2 *87. 6 + 5i( ) 6 − 5i( )
Page 83 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
a + bi and a − bi are called Complex Conjugates. Multiply and simplify, showing all steps. You should get different answers for these. 88. a + b( ) a − b( ) 89. a + bi( ) a − bi( ) Simplify. Show all steps.
90. 2 3 − 3i( ) 2 3 + 3i( )
91. 5 − 3( ) 5 + 3( )
92. 2 3 + −7( ) 2 3 − −7( )
93. 2 3 + 7( ) 2 3 − 7( )
Page 84 of 98 Unit 2 Harland 2.8 Complex Numbers Intermediate Algebra
Simplify. Do not leave any power of i in answer as any power of i can be simplified to i , −i , −1, or 1 . Write your answer as a real number, a pure imaginary number, or as a complex number in the form a + bi or a − bi . Show all steps. BOX answer. These are a bit more challenging.
94. −5 ⋅ −2 ⋅ −3
95. −5 ⋅ −3 ⋅ −2 ⋅ −1 96. 3i ⋅2i2 ⋅5i3
97. 3i3 2i2 − 5i( )
98. 2i − 5i3( ) 2i + 5i3( )
99. 2i2 − 5i6( )2
Page 85 of 98 Unit 2 Harland 2.9 Dividing Complex Numbers Intermediate Algebra
2.9 Dividing Complex Numbers Divide. Simplify. Rationalize all denominators, which means there should not be any imaginary numbers or radicals in the denominator. All radicals should be simplified, with no negative numbers in the radical, and all fractions should be reduced. Do not leave i2 in answer as that can be simplified to -‐1. Write answer as a real number, a pure imaginary number, or as a complex number in the form a + bi or a − bi . Show all steps. BOX answer.
*1. 3i
2. − 5i
3. 32
*4. −52i
5. 2i3
6. 7
5 3i
Page 86 of 98 Unit 2 Harland 2.9 Dividing Complex Numbers Intermediate Algebra
7. 2i−6
*8. 9 − 6i3i
*9. 9 − 6i3
10. 4 −15i−3i
*11. 34i
12. 20 − 5i10i
Page 87 of 98 Unit 2 Harland 2.9 Dividing Complex Numbers Intermediate Algebra
*13. −8 +12i6i
14. 4 − 3i6i
15. 20 − 5i10
16. 10 − 2i−10
Page 88 of 98 Unit 2 Harland 2.9 Dividing Complex Numbers Intermediate Algebra
Complex Conjugates a + bi and a − bi are called complex conjugates. Below is the product of two complex conjugates. a + bi( ) a − bi( ) = a2 − b2i2 = a2 − b2 (−1) = a2 + b2 This essentially is first the difference of two squares, but since there was an i2 , −1 is multiplied by b2 to end up with a2 + b2 for the simplified answer. Remember that a and b represent the real and imaginary parts. b is the coefficient of i. The product, a2 + b2 , is a real number, since there is no i in the answer. This will be used to rationalize denominator that are complex numbers. If both the real and imaginary parts are rational numbers and/or square roots, the product of complex conjugates will not only be real, it will also rational. Note that a2 + b2 is a SUM of two positive numbers since if both a and b are real numbers, then when you square each of them, you get positive numbers. So any time you multiply conjugates, you end up adding two positive numbers. Example: State the conjugate of 5 − 2i . Then multiply 5 − 2i by its conjugate. The conjugate of 5 − 2i is 5 + 2i . If we multiply 5 − 2i by its conjugate, we will get a rational number in the denominator. 5 − 2i( ) 5 + 2i( ) = 25 − 4i2 = 25 − 4(−1) = 25 + 4 = 29 So when we multiply 5 − 2i by its conjugate, we get 29, a rational number. Note that since a + bi( ) a − bi( ) == a2 + b2 , the quick way to multiply two conjugates is to add the squares of the real part and the imaginary part (the coefficient of i). It’s only a sum of two squares if they are complex conjugates! 5 − 2i( ) 5 + 2i( ) == 52 + 22 = 25 + 4 = 29 Be really careful anytime the imaginary part is 1 or -‐1. You won’t usually see the coefficient of 1 or -‐1. Remember that b, the imaginary part is the coefficient of i. Example: State the real part, imaginary part, and conjugate of 2 − i . Then multiply 2 − i its conjugate and simplify. Solution: a = 2 b = −1 Conjugate: 2 + i 2 − i( ) 2 + i( ) = 22 +12 = 4 +1= 5
Page 89 of 98 Unit 2 Harland 2.9 Dividing Complex Numbers Intermediate Algebra
State the real and imaginary part of each complex number. Then state its conjugate. Then multiply each complex number by its conjugate and simplify. Show all steps. 17. 4 + 2i a = ________ b = ________ Conjugate:___________________
Multiply the complex number above by its conjugate and simplify. Show all steps.
18. 7 − 3i a = ________ b = ________ Conjugate:___________________
Multiply the complex number above by its conjugate and simplify. Show all steps.
19. 5 + 2i a = ________ b = ________ Conjugate:___________________
Multiply the complex number above by its conjugate and simplify. Show all steps. 20. 6 + i a = ________ b = ________ Conjugate:___________________
Multiply the complex number above by its conjugate and simplify. Show all steps.
21. 2 6 − 3 2i a = ________ b = ________ Conjugate:___________________ Rationalize the denominator and simplify. Show all steps. Write answer in form a + bi .
*22. 53+ i
Page 90 of 98 Unit 2 Harland 2.9 Dividing Complex Numbers Intermediate Algebra
23. 3
5 − 2i
*24. 103− 4i
25. −2
−1+ 4i
*26. 4i5 + i
Page 91 of 98 Unit 2 Harland 2.9 Dividing Complex Numbers Intermediate Algebra
27. −22i3− 2i
*28. 3+ 2i7 + 2i
29. 5 − i2 − i
30. 3 + 5i5 + 3i
Page 92 of 98 Unit 2 Harland 2.9 Dividing Complex Numbers Intermediate Algebra
Mixed Practice with Complex Numbers
Simplify as much as possible. There should be no negative numbers in square roots, and no irrational or imaginary numbers in the denominator. Write answers as a real number, a pure imaginary number or as a complex number in the form a + bi . Show all steps. BOX answer.
31. −3 −20 + 5i −18 32. 5 −3 ⋅2 −6
33. 45−8
34. 8 − 2i( )− 6 − 3 2i( )
35. 4 − 5i( )2 36. 3 2 − 4i( ) 3 2 + 4i( )
37. i39 38. 3
6 + 2i
Page 93 of 98 Unit 2 Harland 2.10 Review Exercises Intermediate Algebra
2.10 Review Exercises
Answer all problems completely. Show all steps and work. It’s important to be able to do all problems on your own in any order without referring to examples or notes.
Simplify. Assume all variables are positive. All radicals need to be simplified. All denominators needs to be rational, and all fractions need to be reduced.
1. 14x15 ⋅ 21x7
2. −5mn3 108m16n13
3. −3xy −40x11y153
4. 3x 125 − 5 50x2
5. 5x2 x54 − x3 16x4
Page 94 of 98 Unit 2 Harland 2.10 Review Exercises Intermediate Algebra
Simplify. Assume all variables are positive. All radicals need to be simplified. All denominators needs to be rational, and all fractions need to be reduced.
6. 5 10 3 2 − 15( )
7. 2 3 + 3 5( ) 3 3 − 4 5( )
8. 2 3 + 3 5( )2
9. 2 3 + 3 5( ) 2 3 − 3 5( )
10. − 252 15
Page 95 of 98 Unit 2 Harland 2.10 Review Exercises Intermediate Algebra
Simplify. Assume all variables are positive. All radicals need to be simplified. All denominators needs to be rational, and all fractions need to be reduced.
11. −3549x3
12. −12
3 2 + 2
Write each expression using radical notation, and simplify if possible. If there are negative exponents, rewrite the problem using positive exponents before writing in radical notation. If the answer is not a real number, state that. Show all steps.
13. −843
14. 81− 34
Page 96 of 98 Unit 2 Harland 2.10 Review Exercises Intermediate Algebra
Solve each equation, and check each solution. Show all steps, including all steps to checking each solution. PUT a BOX around solution(s) that check.
15. 2x −1 + 2 = 5
16. x +1= 8 − x + 5
17. 3x +1 − x + 4 = 1
Page 97 of 98 Unit 2 Harland 2.10 Review Exercises Intermediate Algebra
Find the exact answer (simplifying any radicals if possible) of the missing side of the right triangle, where a and b represent the legs and c represents the hypotenuse. and then use a calculator to round the answer to one place after the decimal point. Draw a picture of the triangle and Show work. 18. a = 5; b = 11; c = ______ 19. a = 9; c = 13; b = ______ 20. Use the distance formula to compute the exact distance between −1,1( ) and
−3,−7( ) , simplifying any radical if possible. Write the exact answer. Then use a calculator to round to the nearest tenth—one place after decimal point. Show all steps.
Exact answer: __________ Approximation: ____________ Simplify. There should be no powers of i, no negative numbers under a square root, and no square roots or imaginary numbers in the denominator. All radicals and fractions need to be simplified. Show all steps. 21. i43
22. 3 + 2i( )− 5 3 − 4 2i( )
23. −7 ⋅3 −35
Page 98 of 98 Unit 2 Harland 2.10 Review Exercises Intermediate Algebra
Simplify. There should be no powers of i, no negative numbers under a square root, and no square roots or imaginary numbers in the denominator. All radicals and fractions need to be simplified. Show all steps.
24. 4 2 + 3i( ) 4 2 − 3i( )
25. 7 − 3i( )2
26. 3− −15
−6i
27. 2i
4 − 2i