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: ⓐ $d^9$ ⓑ $4^{14x}$ ⓒ $8y^4$ ⓓ $w^6$

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: ⓐ $n^{31}$ ⓑ $3^{x+6}$ ⓒ $56w^6$
ⓓ $a^{16}$

4. ⓐ $q^{27}·q^{15}$ ⓑ $5^x·5^{4x}$ ⓒ $9u^{41}·7u^{53}$
ⓓ $c^5·c^{11}·c^2$

5. $m^x·m^3$

Answer: $m^{x+3}$

6. $n^y·n^2$

7. $y^a·y^b$

Answer: $y^{a+b}$

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: ⓐ $x^{15}$ ⓑ $5^9$ ⓒ $\dfrac{1}{q^{18}}$ ⓓ $\dfrac{1}{10}$

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: ⓐ $p^{14}$ ⓑ $4^{12}$ ⓒ $\dfrac{1}{b^8}$ ⓓ $\dfrac{1}{4^5}$

12. ⓐ $\dfrac{u^{24}}{u^3}$ ⓑ $\dfrac{9^{15}}{9^5}$ ⓒ $\dfrac{x}{x^7}$ ⓓ $\dfrac{10}{10^3}$

13. ⓐ $20^0$ ⓑ $b^0$

Answer: ⓐ 1 ⓑ 1

14. ⓐ $13^0$ ⓑ $k^0$

15. ⓐ $−27^0$ ⓑ $−(27^0)$

Answer: ⓐ $−1$ ⓑ $−1$

16. ⓐ $−15^0$ ⓑ $−(15^0)$

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

17. ⓐ $a^{−2}$ ⓑ $10^{−3}$ ⓒ $\dfrac{1}{c^{−5}}$ ⓓ $\dfrac{1}{3^{−2}}$

Answer: ⓐ $\dfrac{1}{a^{2}}$ ⓑ $\dfrac{1}{1000}$ ⓒ $c^{5}$ ⓓ $9$

18. ⓐ $b^{−4}$ ⓑ $10^{−2}$ ⓒ $\dfrac{1}{c^{−5}}$ ⓓ $\dfrac{1}{5^{−2}}$

19. ⓐ $r^{−3}$ ⓑ $10^{−5}$ ⓒ $\dfrac{1}{q^{−10}}$ ⓓ $\dfrac{1}{10^{−3}}$

Answer: ⓐ $\dfrac{1}{r3}$ ⓑ $\dfrac{1}{100,000}$ ⓒ $q^{10}$ ⓓ $1,000$

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: ⓐ $\dfrac{64}{25}$ ⓑ $\dfrac{a^{2}}{b^{2}}$

22. ⓐ $\left(\dfrac{3}{10}\right)^{−2}$ ⓑ $\left(−\dfrac{2}{z}\right)^{−3}$

23. ⓐ $\left(\dfrac{4}{9}\right)^{−3}$ ⓑ $\left(−\dfrac{u}{v}\right)^{−5}$

Answer: ⓐ $\dfrac{729}{64}$ ⓑ $−\dfrac{v^{5}}{u^{5}}$

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: ⓐ $\dfrac{1}{25}$ ⓑ $−\dfrac{1}{25}$ ⓒ $25$ ⓓ $−25$

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: ⓐ $\dfrac{3}{5}$ ⓑ $\dfrac{1}{15}$

28. ⓐ $3·4^{−2}$ ⓑ $(3·4)^{−2}$

In the following exercises, simplify each expression using the Product Property.

29. ⓐ $b^{4}b^{−8}$ ⓑ $(w^{4}x^{−5})(w^{−2}x^{−4})$) ⓒ $(−6c^{−3}d^9)(2c^4d^{−5})$

Answer: ⓐ $\dfrac{1}{b^{4}}$ ⓑ $\dfrac{w^{2}}{x^{9}}$ ⓒ $−12cd^{4}$

30. ⓐ $s^{3}·s^{−7}$ ⓑ $(m^{3}n^{−3})(m^{5}n^{−1})$
ⓒ $(−2j^{−5}k^{8})(7j^{2}k^{−3})$

31. ⓐ $a^{3}·a^{−3}$ ⓑ $(uv^{−2})(u^{−5}v^{−3})$
ⓒ $(−4r^{−2}s^{−8})(9r^{4}s^{3})$

Answer: ⓐ $1$ ⓑ $\dfrac{1}{u^{4}v^{5}}$ ⓒ $−36\dfrac{r^{2}}{j^{5}}$

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: $\dfrac{1}{p}$

34. $x^{4}·x^{−2}·x^{−3}$

In the following exercises, simplify each expression using the Power Property.

35. ⓐ $(m^4)^2$ ⓑ $(10^3)^6$ ⓒ $(x^3)^{−4}$

Answer: ⓐ $m^{8}$ ⓑ $10^{18}$ ⓒ $\dfrac{1}{x^{12}}$

36. ⓐ $(b^{2})^{7}$ ⓑ $(3^8)^2$ ⓒ $(k^2)^{−5}$

37. ⓐ $(y^3)^x$ ⓑ $(5^x)^x$ ⓒ $(q^6)^{−8}$

Answer: ⓐ $y^{3x}$ ⓑ $5^{xy}$ ⓒ $\dfrac{1}{q^{48}}$

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. ⓐ $(−3xy)^2$ ⓑ $(6a)^0$ ⓒ $(5x^2)^{−2}$ ⓓ $(−4y^{−3})^2$

Answer: ⓐ $9x^2y^2$ ⓑ 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: ⓐ $−125a^3b^3$ ⓑ 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: ⓐ $\dfrac{p^5}{32}$ ⓑ $\dfrac{y^6}{x^6}$ ⓒ $\dfrac{8x^3y^6}{z^3}$
ⓓ $\dfrac{16}{p^6q^4}$

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: ⓐ $\dfrac{a^4}{81b^4}$ ⓑ $\dfrac{16m^2}{25}$ ⓒ $\dfrac{a^4c^4}{9b^6}$ ⓓ $\dfrac{q^8r^8}{p^2}$

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. ⓐ $(5t^2)^3(3t)^2$ ⓑ $\dfrac{(t^2)^5(t^{−4})^2}{(t^3)^7}$ ⓒ $\left(\dfrac{2xy^2}{x^3y^{−2}}\right)^2\left(\dfrac{12xy^3}{x^3y^{−1}}\right)^{−1}$

Answer: ⓐ $1125t^8$ ⓑ $\dfrac{1}{t^{19}}$ ⓒ $\dfrac{y^4}{3x^2}$

48. ⓐ $(10k^4)^3(5k^6)^2$ ⓑ $\dfrac{(q^3)^6(q^{−2})^3}{(q^4)^8}$

49. ⓐ $(m^2n)^2(2mn^5)^4$ ⓑ $\dfrac{(−2p^{−2})^4(3p^4)^2}{(−6p^3)^2}$

Answer: ⓐ $16m^8n^{22}$ ⓑ $\dfrac{4}{p^6}$

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. ⓐ $7n^{−1}$ ⓑ $(7n)^{−1}$ ⓒ $(−7n)^{−1}$

Answer: ⓐ $\dfrac{7}{n}$ ⓑ $\dfrac{1}{7n}$ ⓒ $−\dfrac{1}{7n}$

52. ⓐ $6r^{−1}$ ⓑ $(6r)^{−1}$ ⓒ $(−6r)^{−1}$

53. ⓐ $(3p)^{−2}$ ⓑ $3p^{−2}$ ⓒ $−3p^{−2}$

Answer: ⓐ $\dfrac{1}{9p^2}$ ⓑ $\dfrac{3}{p^2}$ ⓒ $−\dfrac{3}{p^2}$

54. ⓐ $(2q)^{−4}$ ⓑ $2q^{−4}$ ⓒ $−2q^{−4}$

55. $(x^2)^4·(x^3)^2$

Answer: $x^{14}$

56. $(y^4)^3·(y^5)^2$

57. $(a^2)^6·(a^3)^8$

Answer: $a^{30}$

58. $(b^7)^5·(b^2)^6$

59. $(2m^6)^3$

Answer: $2m^{18}$

60. $(3y^2)^4$

61. $(10x^2y)^3$

Answer: $1,000x^6y^3$

62. $(2mn^4)^5$

63. $(−2a^3b^2)^4$

Answer: $16a^{12}b^8$

64. $(−10u^2v^4)^3$

65. $\left(\dfrac{2}{3}x^2y\right)^3$

Answer: $\dfrac{8}{27}x^6y^3$

66. $\left(\dfrac{7}{9}pq^4\right)^2$

67. $(8a^3)^2(2a)^4$

Answer: $1,024a^{10}$

68. $(5r^2)^3(3r)^2$

69. $(10p^4)^3(5p^6)^2$

Answer: $25,000p^{24}$

70. $(4x^3)^3(2x^5)^4$

71. $\left(\dfrac{1}{2}x^2y^3\right)^4\left(4x^5y^3\right)^2$

Answer: $x^{18}y^{18}$

72. $\left(\dfrac{1}{3}m^3n^2\right)^4\left(9m^8n^3\right)^2$

73. $(3m^2n)^2(2mn^5)^4$

Answer: $144m^8n^{22}$

74. $(2pq^4)^3(5p^6q)^2$

75. ⓐ $(3x)^2(5x)$ ⓑ $(2y)^3(6y)$

Answer: ⓐ $45x^3$ ⓑ $48y^4$

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. ⓐ $(2r^{−2})^3(4^{−1}r)^2$ ⓑ $(3x^{−3})^3(3^{−1}x^5)^4$

Answer: ⓐ $12r^4$ ⓑ $13x^{11}$

78. $\left(\dfrac{k^{−2}k^8}{k^3}\right)^2$

79. $\left(\dfrac{j^{−2}j^5}{j^4}\right)^3$

Answer: $\dfrac{1}{j^3}$

80. $\dfrac{(−4m^{−3})^2(5m^4)^3}{(−10m^6)^3}$

81. $\dfrac{(−10n^{−2})^3(4n^5)^2}{(2n^8)^2}$

Answer: $−\dfrac{4000}{n^{12}}$

Use Scientific Notation

In the following exercises, write each number in scientific notation.

82. ⓐ 57,000 ⓑ 0.026

83. ⓐ 340,000 ⓑ 0.041

Answer: ⓐ $34\times10^4$ ⓑ $41\times10^{−3}$

84. ⓐ 8,750,000 ⓑ 0.00000871

85. ⓐ 1,290,000 ⓑ 0.00000103

Answer

ⓐ $1.29\times10^6$

ⓑ $103\times10^{−8}$

In the following exercises, convert each number to decimal form.

86. ⓐ $5.2\times10^2$ ⓑ $2.5\times10^{−2}$

87. ⓐ $−8.3\times10^2$ ⓑ $3.8\times10^{−2}$

Answer: ⓐ $−830$ ⓑ 0.038

88. ⓐ $7.5\times10^6$ ⓑ $−4.13\times10^{−5}$

89. ⓐ $1.6\times10^{10}$ ⓑ $8.43\times10^{−6}$

Answer: ⓐ 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. ⓐ $(2\times10^2)(1\times10^{−4})$ ⓑ $\dfrac{5\times10^{−2}}{1\times10^{−10}}$

Answer: ⓐ 0.02 ⓑ 500,000,000

92. ⓐ $(7.1\times10^{−2})(2.4\times10^{−4})$ ⓑ $\dfrac{6\times10^4}{3\times10^{−2}}$

93. ⓐ $(3.5\times10^{−4})(1.6\times10^{−2})$ ⓑ $\dfrac{8\times10^6}{4\times10^{−1}}$

Answer: ⓐ 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?

Answer: 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: Answers will vary.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

$This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions using the properties for exponents.”, “use the definition of a negative exponent”, and “use scientific notation”. The other columns are left blank so that the learner may indicate their mastery level for each topic.$

ⓑ After reviewing this checklist, what will you do to become confident for all goals?