5.3E: Exercises
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Practice Makes Perfect
Simplify Expressions Using the Properties for Exponents
In the following exercises, simplify each expression using the properties for exponents.
1. ⓐ d3·d6 ⓑ 45x·49x ⓒ 2y·4y3 ⓓ w·w2·w3
- Answer
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ⓐ d9 ⓑ 414x ⓒ 8y4 ⓓ w6
2. ⓐ x4·x2 ⓑ 89x·83 ⓒ 3z25·5z8 ⓓ y·y3·y5
3. ⓐ n19·n12 ⓑ 3x·36 ⓒ 7w5·8w ⓓ a4·a3·a9
- Answer
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ⓐ n31 ⓑ 3x+6 ⓒ 56w6
ⓓ a16
4. ⓐ q27·q15 ⓑ 5x·54x ⓒ 9u41·7u53
ⓓ c5·c11·c2
5. mx·m3
- Answer
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mx+3
6. ny·n2
7. ya·yb
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ya+b
8. xp·xq
9. ⓐ x18x3 ⓑ 51253 ⓒ q18q36 ⓓ 102103
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ⓐ x15 ⓑ 59 ⓒ 1q18 ⓓ 110
10. ⓐ y20y10 ⓑ 71672 ⓒ t10t40 ⓓ 8385
11. ⓐ p21p7 ⓑ 41644 ⓒ bb9 ⓓ 446
- Answer
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ⓐ p14 ⓑ 412 ⓒ 1b8 ⓓ 145
12. ⓐ u24u3 ⓑ 91595 ⓒ xx7 ⓓ 10103
13. ⓐ 200 ⓑ b0
- Answer
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ⓐ 1 ⓑ 1
14. ⓐ 130 ⓑ k0
15. ⓐ −270 ⓑ −(270)
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ⓐ −1 ⓑ −1
16. ⓐ −150 ⓑ −(150)
Use the Definition of a Negative Exponent
In the following exercises, simplify each expression.
17. ⓐ a−2 ⓑ 10−3 ⓒ 1c−5 ⓓ 13−2
- Answer
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ⓐ 1a2 ⓑ 11000 ⓒ c5 ⓓ 9
18. ⓐ b−4 ⓑ 10−2 ⓒ 1c−5 ⓓ 15−2
19. ⓐ r−3 ⓑ 10−5 ⓒ 1q−10 ⓓ 110−3
- Answer
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ⓐ 1r3 ⓑ 1100,000 ⓒ q10 ⓓ 1,000
20. ⓐ s−8 ⓑ 10−2 ⓒ 1t−9 ⓓ 110−4
21. ⓐ (58)−2 ⓑ (−ba)−2
- Answer
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ⓐ 6425 ⓑ a2b2
22. ⓐ (310)−2 ⓑ (−2z)−3
23. ⓐ (49)−3 ⓑ (−uv)−5
- Answer
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ⓐ 72964 ⓑ −v5u5
24. ⓐ (72)−3 ⓑ (−3x)−3
25. ⓐ (−5)−2 ⓑ −5−2 ⓒ (−15)−2 ⓓ −(15)−2
- Answer
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ⓐ 125 ⓑ −125 ⓒ 25 ⓓ −25
26. ⓐ −5−3 ⓑ (−15)−3 ⓒ −(15)−3 ⓓ (−5)−3
27. ⓐ 3·5−1 ⓑ (3·5)−1
- Answer
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ⓐ 35 ⓑ 115
28. ⓐ 3·4−2 ⓑ (3·4)−2
In the following exercises, simplify each expression using the Product Property.
29. ⓐ b4b−8 ⓑ (w4x−5)(w−2x−4)) ⓒ (−6c−3d9)(2c4d−5)
- Answer
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ⓐ 1b4 ⓑ w2x9 ⓒ −12cd4
30. ⓐ s3·s−7 ⓑ (m3n−3)(m5n−1)
ⓒ (−2j−5k8)(7j2k−3)
31. ⓐ a3·a−3 ⓑ (uv−2)(u−5v−3)
ⓒ (−4r−2s−8)(9r4s3)
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ⓐ 1 ⓑ 1u4v5 ⓒ −36r2j5
32. ⓐ y5·y−5 ⓑ (pq−4)(p−6q−3)
ⓒ (−5m4n6)(8m−5n−3)
33. p5·p−2·p−4
- Answer
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1p
34. x4·x−2·x−3
In the following exercises, simplify each expression using the Power Property.
35. ⓐ (m4)2 ⓑ (103)6 ⓒ (x3)−4
- Answer
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ⓐ m8 ⓑ 1018 ⓒ 1x12
36. ⓐ (b2)7 ⓑ (38)2 ⓒ (k2)−5
37. ⓐ (y3)x ⓑ (5x)x ⓒ (q6)−8
- Answer
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ⓐ y3x ⓑ 5xy ⓒ 1q48
38. ⓐ (x2)y ⓑ (7a)b ⓒ (a9)−10
In the following exercises, simplify each expression using the Product to a Power Property.
39. ⓐ (−3xy)2 ⓑ (6a)0 ⓒ (5x2)−2 ⓓ (−4y−3)2
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ⓐ 9x2y2 ⓑ 1 ⓒ 125x4 ⓓ 16y6
40. ⓐ (−4ab)2 ⓑ (5x)0 ⓒ (4y3)−3 ⓓ (−7y−3)2
41. ⓐ (−5ab)3 ⓑ (−4pq)0 ⓒ (−6x3)−2 ⓓ (3y−4)2
- Answer
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ⓐ −125a3b3 ⓑ 1 ⓒ 136x6 ⓓ 9y8
42. ⓐ (−3xyz)4 ⓑ (−7mn)0 ⓒ (−3x3)−2
ⓓ (2y−5)2
In the following exercises, simplify each expression using the Quotient to a Power Property.
43. ⓐ (p2)5 ⓑ (xy)−6 ⓒ (2xy2z)3 ⓓ (4p−3q2)2
- Answer
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ⓐ p532 ⓑ y6x6 ⓒ 8x3y6z3
ⓓ 16p6q4
44. ⓐ (x3)4 ⓑ (ab)−5 ⓒ (2xy2z)3 ⓓ (x3yz4)2
45. ⓐ (a3b)4 ⓑ (54m)−2 ⓒ (3a−2b3c3)−2 ⓓ (p−1q4r−4)2
- Answer
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ⓐ a481b4 ⓑ 16m225 ⓒ a4c49b6 ⓓ q8r8p2
46. ⓐ (x2y)3 ⓑ (103q)−4 ⓒ (2x3y43z2)5 ⓓ (5a3b−12c4)−3
In the following exercises, simplify each expression by applying several properties.
47. ⓐ (5t2)3(3t)2 ⓑ (t2)5(t−4)2(t3)7 ⓒ (2xy2x3y−2)2(12xy3x3y−1)−1
- Answer
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ⓐ 1125t8 ⓑ 1t19 ⓒ y43x2
48. ⓐ (10k4)3(5k6)2 ⓑ (q3)6(q−2)3(q4)8
49. ⓐ (m2n)2(2mn5)4 ⓑ (−2p−2)4(3p4)2(−6p3)2
- Answer
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ⓐ 16m8n22 ⓑ 4p6
50. ⓐ (3pq4)2(6p6q)2 ⓑ (−2k−3)2(6k2)4(9k4)2
Mixed Practice
In the following exercises, simplify each expression.
51. ⓐ 7n−1 ⓑ (7n)−1 ⓒ (−7n)−1
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ⓐ 7n ⓑ 17n ⓒ −17n
52. ⓐ 6r−1 ⓑ (6r)−1 ⓒ (−6r)−1
53. ⓐ (3p)−2 ⓑ 3p−2 ⓒ −3p−2
- Answer
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ⓐ 19p2 ⓑ 3p2 ⓒ −3p2
54. ⓐ (2q)−4 ⓑ 2q−4 ⓒ −2q−4
55. (x2)4·(x3)2
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x14
56. (y4)3·(y5)2
57. (a2)6·(a3)8
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a30
58. (b7)5·(b2)6
59. (2m6)3
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2m18
60. (3y2)4
61. (10x2y)3
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1,000x6y3
62. (2mn4)5
63. (−2a3b2)4
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16a12b8
64. (−10u2v4)3
65. (23x2y)3
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827x6y3
66. (79pq4)2
67. (8a3)2(2a)4
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1,024a10
68. (5r2)3(3r)2
69. (10p4)3(5p6)2
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25,000p24
70. (4x3)3(2x5)4
71. (12x2y3)4(4x5y3)2
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x18y18
72. (13m3n2)4(9m8n3)2
73. (3m2n)2(2mn5)4
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144m8n22
74. (2pq4)3(5p6q)2
75. ⓐ (3x)2(5x) ⓑ (2y)3(6y)
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ⓐ 45x3 ⓑ 48y4
76. ⓐ (12y2)3(23y)2 ⓑ (12j2)5(25j3)2
77. ⓐ (2r−2)3(4−1r)2 ⓑ (3x−3)3(3−1x5)4
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ⓐ 12r4 ⓑ 13x11
78. (k−2k8k3)2
79. (j−2j5j4)3
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1j3
80. (−4m−3)2(5m4)3(−10m6)3
81. (−10n−2)3(4n5)2(2n8)2
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−4000n12
Use Scientific Notation
In the following exercises, write each number in scientific notation.
82. ⓐ 57,000 ⓑ 0.026
83. ⓐ 340,000 ⓑ 0.041
- Answer
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ⓐ 34×104 ⓑ 41×10−3
84. ⓐ 8,750,000 ⓑ 0.00000871
85. ⓐ 1,290,000 ⓑ 0.00000103
- Answer
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ⓐ 1.29×106
ⓑ 103×10−8
In the following exercises, convert each number to decimal form.
86. ⓐ 5.2×102 ⓑ 2.5×10−2
87. ⓐ −8.3×102 ⓑ 3.8×10−2
- Answer
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ⓐ −830 ⓑ 0.038
88. ⓐ 7.5×106 ⓑ −4.13×10−5
89. ⓐ 1.6×1010 ⓑ 8.43×10−6
- Answer
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ⓐ 16,000,000,000
ⓑ 0.00000843
In the following exercises, multiply or divide as indicated. Write your answer in decimal form.
90. ⓐ (3×10−5)(3×109) ⓑ 7×10−31×10−7
91. ⓐ (2×102)(1×10−4) ⓑ 5×10−21×10−10
- Answer
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ⓐ 0.02 ⓑ 500,000,000
92. ⓐ (7.1×10−2)(2.4×10−4) ⓑ 6×1043×10−2
93. ⓐ (3.5×10−4)(1.6×10−2) ⓑ 8×1064×10−1
- Answer
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ⓐ 0.0000056 ⓑ 20,000,000
Writing Exercises
94. Use the Product Property for Exponents to explain why x·x=x2.
95. Jennifer thinks the quotient a24a6 simplifies to a4. What is wrong with her reasoning?
- Answer
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Answers will vary.
96. Explain why −53=(−5)3 but −54≠(−5)4.
97. When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?
- Answer
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Answers will vary.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all goals?