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. ⓐ d^3·d^6 ⓑ 4^{5x}·4^{9x} ⓒ 2y·4y^3 ⓓ w·w^2·w^3
- Answer
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ⓐ d^9 ⓑ 4^{14x} ⓒ 8y^4 ⓓ w^6
2. ⓐ x^4·x^2 ⓑ 8^{9x}·8^3 ⓒ 3z^{25}·5z^8 ⓓ y·y^3·y^5
3. ⓐ n^{19}·n^{12} ⓑ 3^x·3^6 ⓒ 7w^5·8w ⓓ a^4·a^3·a^9
- Answer
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ⓐ n^{31} ⓑ 3^{x+6} ⓒ 56w^6
ⓓ a^{16}
4. ⓐ q^{27}·q^{15} ⓑ 5^x·5^{4x} ⓒ 9u^{41}·7u^{53}
ⓓ c^5·c^{11}·c^2
5. m^x·m^3
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m^{x+3}
6. n^y·n^2
7. y^a·y^b
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y^{a+b}
8. x^p·x^q
9. ⓐ \dfrac{x^{18}}{x^3} ⓑ \dfrac{5^{12}}{5^3} ⓒ \dfrac{q^{18}}{q^{36}} ⓓ \dfrac{10^2}{10^3}
- Answer
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ⓐ x^{15} ⓑ 5^9 ⓒ \dfrac{1}{q^{18}} ⓓ \dfrac{1}{10}
10. ⓐ \dfrac{y^{20}}{y^{10}} ⓑ \dfrac{7^{16}}{7^2} ⓒ \dfrac{t^{10}}{t^{40}} ⓓ \dfrac{8^3}{8^5}
11. ⓐ \dfrac{p^{21}}{p^7} ⓑ \dfrac{4^{16}}{4^4} ⓒ \dfrac{b}{b^9} ⓓ \dfrac{4}{4^6}
- Answer
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ⓐ p^{14} ⓑ 4^{12} ⓒ \dfrac{1}{b^8} ⓓ \dfrac{1}{4^5}
12. ⓐ \dfrac{u^{24}}{u^3} ⓑ \dfrac{9^{15}}{9^5} ⓒ \dfrac{x}{x^7} ⓓ \dfrac{10}{10^3}
13. ⓐ 20^0 ⓑ b^0
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ⓐ 1 ⓑ 1
14. ⓐ 13^0 ⓑ k^0
15. ⓐ −27^0 ⓑ −(27^0)
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ⓐ −1 ⓑ −1
16. ⓐ −15^0 ⓑ −(15^0)
Use the Definition of a Negative Exponent
In the following exercises, simplify each expression.
17. ⓐ a^{−2} ⓑ 10^{−3} ⓒ \dfrac{1}{c^{−5}} ⓓ \dfrac{1}{3^{−2}}
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ⓐ \dfrac{1}{a^{2}} ⓑ \dfrac{1}{1000} ⓒ c^{5} ⓓ 9
18. ⓐ b^{−4} ⓑ 10^{−2} ⓒ \dfrac{1}{c^{−5}} ⓓ \dfrac{1}{5^{−2}}
19. ⓐ r^{−3} ⓑ 10^{−5} ⓒ \dfrac{1}{q^{−10}} ⓓ \dfrac{1}{10^{−3}}
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ⓐ \dfrac{1}{r3} ⓑ \dfrac{1}{100,000} ⓒ q^{10} ⓓ 1,000
20. ⓐ s^{−8} ⓑ 10^{−2} ⓒ \dfrac{1}{t^{−9}} ⓓ \dfrac{1}{10^{−4}}
21. ⓐ \left(\dfrac{5}{8}\right)^{-2} ⓑ \left(−\dfrac{b}{a}\right)^{−2}
- Answer
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ⓐ \dfrac{64}{25} ⓑ \dfrac{a^{2}}{b^{2}}
22. ⓐ \left(\dfrac{3}{10}\right)^{−2} ⓑ \left(−\dfrac{2}{z}\right)^{−3}
23. ⓐ \left(\dfrac{4}{9}\right)^{−3} ⓑ \left(−\dfrac{u}{v}\right)^{−5}
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ⓐ \dfrac{729}{64} ⓑ −\dfrac{v^{5}}{u^{5}}
24. ⓐ \left(\dfrac{7}{2}\right)^{−3} ⓑ \left(−\dfrac{3}{x}\right)^{−3}
25. ⓐ (−5)^{−2} ⓑ −5^{−2} ⓒ \left(−\dfrac{1}{5}\right)^{−2} ⓓ −\left(\dfrac{1}{5}\right)^{−2}
- Answer
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ⓐ \dfrac{1}{25} ⓑ −\dfrac{1}{25} ⓒ 25 ⓓ −25
26. ⓐ −5^{−3} ⓑ \left(−\dfrac{1}{5}\right)^{−3} ⓒ −\left(\dfrac{1}{5}\right)^{−3} ⓓ (−5)^{−3}
27. ⓐ 3·5^{−1} ⓑ (3·5)^{−1}
- Answer
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ⓐ \dfrac{3}{5} ⓑ \dfrac{1}{15}
28. ⓐ 3·4^{−2} ⓑ (3·4)^{−2}
In the following exercises, simplify each expression using the Product Property.
29. ⓐ b^{4}b^{−8} ⓑ (w^{4}x^{−5})(w^{−2}x^{−4})) ⓒ (−6c^{−3}d^9)(2c^4d^{−5})
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ⓐ \dfrac{1}{b^{4}} ⓑ \dfrac{w^{2}}{x^{9}} ⓒ −12cd^{4}
30. ⓐ s^{3}·s^{−7} ⓑ (m^{3}n^{−3})(m^{5}n^{−1})
ⓒ (−2j^{−5}k^{8})(7j^{2}k^{−3})
31. ⓐ a^{3}·a^{−3} ⓑ (uv^{−2})(u^{−5}v^{−3})
ⓒ (−4r^{−2}s^{−8})(9r^{4}s^{3})
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ⓐ 1 ⓑ \dfrac{1}{u^{4}v^{5}} ⓒ −36\dfrac{r^{2}}{j^{5}}
32. ⓐ y^{5}·y^{−5} ⓑ (pq^{−4})(p^{−6}q^{−3})
ⓒ (−5m^{4}n^{6})(8m^{−5}n^{−3})
33. p^{5}·p^{−2}·p^{−4}
- Answer
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\dfrac{1}{p}
34. x^{4}·x^{−2}·x^{−3}
In the following exercises, simplify each expression using the Power Property.
35. ⓐ (m^4)^2 ⓑ (10^3)^6 ⓒ (x^3)^{−4}
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ⓐ m^{8} ⓑ 10^{18} ⓒ \dfrac{1}{x^{12}}
36. ⓐ (b^{2})^{7} ⓑ (3^8)^2 ⓒ (k^2)^{−5}
37. ⓐ (y^3)^x ⓑ (5^x)^x ⓒ (q^6)^{−8}
- Answer
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ⓐ y^{3x} ⓑ 5^{xy} ⓒ \dfrac{1}{q^{48}}
38. ⓐ (x^2)^y ⓑ (7^a)^b ⓒ (a^9)^{−10}
In the following exercises, simplify each expression using the Product to a Power Property.
39. ⓐ (−3xy)^2 ⓑ (6a)^0 ⓒ (5x^2)^{−2} ⓓ (−4y^{−3})^2
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ⓐ 9x^2y^2 ⓑ 1 ⓒ \dfrac{1}{25x^4} ⓓ \dfrac{16}{y^6}
40. ⓐ (−4ab)^2 ⓑ (5x)^0 ⓒ (4y^3)^{−3} ⓓ (−7y^{−3})^2
41. ⓐ (−5ab)^3 ⓑ (−4pq)^0 ⓒ (−6x^3)^{−2} ⓓ (3y^{−4})^2
- Answer
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ⓐ −125a^3b^3 ⓑ 1 ⓒ \dfrac{1}{36x^6} ⓓ \dfrac{9}{y^8}
42. ⓐ (−3xyz)^4 ⓑ (−7mn)^0 ⓒ (−3x^3)^{−2}
ⓓ (2y^{−5})^2
In the following exercises, simplify each expression using the Quotient to a Power Property.
43. ⓐ (p^2)^5 ⓑ \left(\dfrac{x}{y}\right)^{−6} ⓒ \left(\dfrac{2xy^2}{z}\right)^3 ⓓ \left(\dfrac{4p^{−3}}{q^2}\right)^2
- Answer
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ⓐ \dfrac{p^5}{32} ⓑ \dfrac{y^6}{x^6} ⓒ \dfrac{8x^3y^6}{z^3}
ⓓ \dfrac{16}{p^6q^4}
44. ⓐ \left(\dfrac{x}{3}\right)^4 ⓑ \left(\dfrac{a}{b}\right)^{−5} ⓒ \left(\dfrac{2xy^2}{z}\right)^3 ⓓ \left(\dfrac{x^3y}{z^4}\right)^2
45. ⓐ \left(\dfrac{a}{3b}\right)^4 ⓑ \left(\dfrac{5}{4m}\right)^{−2} ⓒ \left(\dfrac{3a^{−2}b^3}{c^3}\right)^{−2} ⓓ \left(\dfrac{p^{−1}q^4}{r^{−4}}\right)^2
- Answer
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ⓐ \dfrac{a^4}{81b^4} ⓑ \dfrac{16m^2}{25} ⓒ \dfrac{a^4c^4}{9b^6} ⓓ \dfrac{q^8r^8}{p^2}
46. ⓐ \left(\dfrac{x^2}{y}\right)^3 ⓑ \left(\dfrac{10}{3q}\right)^{−4} ⓒ \left(\dfrac{2x^3y^4}{3z^2}\right)^5 ⓓ \left(\dfrac{5a^3b^{−1}}{2c^4}\right)^{−3}
In the following exercises, simplify each expression by applying several properties.
47. ⓐ (5t^2)^3(3t)^2 ⓑ \dfrac{(t^2)^5(t^{−4})^2}{(t^3)^7} ⓒ \left(\dfrac{2xy^2}{x^3y^{−2}}\right)^2\left(\dfrac{12xy^3}{x^3y^{−1}}\right)^{−1}
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ⓐ 1125t^8 ⓑ \dfrac{1}{t^{19}} ⓒ \dfrac{y^4}{3x^2}
48. ⓐ (10k^4)^3(5k^6)^2 ⓑ \dfrac{(q^3)^6(q^{−2})^3}{(q^4)^8}
49. ⓐ (m^2n)^2(2mn^5)^4 ⓑ \dfrac{(−2p^{−2})^4(3p^4)^2}{(−6p^3)^2}
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ⓐ 16m^8n^{22} ⓑ \dfrac{4}{p^6}
50. ⓐ (3pq^4)^2(6p^6q)^2 ⓑ \dfrac{(−2k^{−3})^2(6k^2)^4}{(9k^4)^2}
Mixed Practice
In the following exercises, simplify each expression.
51. ⓐ 7n^{−1} ⓑ (7n)^{−1} ⓒ (−7n)^{−1}
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ⓐ \dfrac{7}{n} ⓑ \dfrac{1}{7n} ⓒ −\dfrac{1}{7n}
52. ⓐ 6r^{−1} ⓑ (6r)^{−1} ⓒ (−6r)^{−1}
53. ⓐ (3p)^{−2} ⓑ 3p^{−2} ⓒ −3p^{−2}
- Answer
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ⓐ \dfrac{1}{9p^2} ⓑ \dfrac{3}{p^2} ⓒ −\dfrac{3}{p^2}
54. ⓐ (2q)^{−4} ⓑ 2q^{−4} ⓒ −2q^{−4}
55. (x^2)^4·(x^3)^2
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x^{14}
56. (y^4)^3·(y^5)^2
57. (a^2)^6·(a^3)^8
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a^{30}
58. (b^7)^5·(b^2)^6
59. (2m^6)^3
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2m^{18}
60. (3y^2)^4
61. (10x^2y)^3
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1,000x^6y^3
62. (2mn^4)^5
63. (−2a^3b^2)^4
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16a^{12}b^8
64. (−10u^2v^4)^3
65. \left(\dfrac{2}{3}x^2y\right)^3
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\dfrac{8}{27}x^6y^3
66. \left(\dfrac{7}{9}pq^4\right)^2
67. (8a^3)^2(2a)^4
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1,024a^{10}
68. (5r^2)^3(3r)^2
69. (10p^4)^3(5p^6)^2
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25,000p^{24}
70. (4x^3)^3(2x^5)^4
71. \left(\dfrac{1}{2}x^2y^3\right)^4\left(4x^5y^3\right)^2
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x^{18}y^{18}
72. \left(\dfrac{1}{3}m^3n^2\right)^4\left(9m^8n^3\right)^2
73. (3m^2n)^2(2mn^5)^4
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144m^8n^{22}
74. (2pq^4)^3(5p^6q)^2
75. ⓐ (3x)^2(5x) ⓑ (2y)^3(6y)
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ⓐ 45x^3 ⓑ 48y^4
76. ⓐ \left(\dfrac{1}{2}y^2\right)^3\left(\dfrac{2}{3}y\right)^2 ⓑ \left(\dfrac{1}{2}j^2\right)^5\left(\dfrac{2}{5}j^3\right)^2
77. ⓐ (2r^{−2})^3(4^{−1}r)^2 ⓑ (3x^{−3})^3(3^{−1}x^5)^4
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ⓐ 12r^4 ⓑ 13x^{11}
78. \left(\dfrac{k^{−2}k^8}{k^3}\right)^2
79. \left(\dfrac{j^{−2}j^5}{j^4}\right)^3
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\dfrac{1}{j^3}
80. \dfrac{(−4m^{−3})^2(5m^4)^3}{(−10m^6)^3}
81. \dfrac{(−10n^{−2})^3(4n^5)^2}{(2n^8)^2}
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−\dfrac{4000}{n^{12}}
Use Scientific Notation
In the following exercises, write each number in scientific notation.
82. ⓐ 57,000 ⓑ 0.026
83. ⓐ 340,000 ⓑ 0.041
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ⓐ 34\times10^4 ⓑ 41\times10^{−3}
84. ⓐ 8,750,000 ⓑ 0.00000871
85. ⓐ 1,290,000 ⓑ 0.00000103
- Answer
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ⓐ 1.29\times10^6
ⓑ 103\times10^{−8}
In the following exercises, convert each number to decimal form.
86. ⓐ 5.2\times10^2 ⓑ 2.5\times10^{−2}
87. ⓐ −8.3\times10^2 ⓑ 3.8\times10^{−2}
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ⓐ −830 ⓑ 0.038
88. ⓐ 7.5\times10^6 ⓑ −4.13\times10^{−5}
89. ⓐ 1.6\times10^{10} ⓑ 8.43\times10^{−6}
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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\times10^{−5})(3\times10^9) ⓑ \dfrac{7\times10^{−3}}{1\times10^{−7}}
91. ⓐ (2\times10^2)(1\times10^{−4}) ⓑ \dfrac{5\times10^{−2}}{1\times10^{−10}}
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ⓐ 0.02 ⓑ 500,000,000
92. ⓐ (7.1\times10^{−2})(2.4\times10^{−4}) ⓑ \dfrac{6\times10^4}{3\times10^{−2}}
93. ⓐ (3.5\times10^{−4})(1.6\times10^{−2}) ⓑ \dfrac{8\times10^6}{4\times10^{−1}}
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ⓐ 0.0000056 ⓑ 20,000,000
Writing Exercises
94. Use the Product Property for Exponents to explain why x·x=x^2.
95. Jennifer thinks the quotient \dfrac{a^{24}}{a^6} simplifies to a^4. What is wrong with her reasoning?
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Answers will vary.
96. Explain why −5^3=(−5)^3 but −5^4 \neq (−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?