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# 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. ⓐ $$d^3·d^6$$ ⓑ $$4^{5x}·4^{9x}$$ ⓒ $$2y·4y^3$$ ⓓ $$w·w^2·w^3$$

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

2. ⓐ $$x^4·x^2$$ ⓑ $$8^{9x}·8^3$$ ⓒ $$3z^{25}·5z^8$$ ⓓ $$y·y^3·y^5$$

3. ⓐ $$n^{19}·n^{12}$$ ⓑ $$3^x·3^6$$ ⓒ $$7w^5·8w$$ ⓓ $$a^4·a^3·a^9$$

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

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

5. $$m^x·m^3$$

$$m^{x+3}$$

6. $$n^y·n^2$$

7. $$y^a·y^b$$

$$y^{a+b}$$

8. $$x^p·x^q$$

9. ⓐ $$\dfrac{x^{18}}{x^3}$$ ⓑ $$\dfrac{5^{12}}{5^3}$$ ⓒ $$\dfrac{q^{18}}{q^{36}}$$ ⓓ $$\dfrac{10^2}{10^3}$$

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

10. ⓐ $$\dfrac{y^{20}}{y^{10}}$$ ⓑ $$\dfrac{7^{16}}{7^2}$$ ⓒ $$\dfrac{t^{10}}{t^{40}}$$ ⓓ $$\dfrac{8^3}{8^5}$$

11. ⓐ $$\dfrac{p^{21}}{p^7}$$ ⓑ $$\dfrac{4^{16}}{4^4}$$ ⓒ $$\dfrac{b}{b^9}$$ ⓓ $$\dfrac{4}{4^6}$$

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

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

13. ⓐ $$20^0$$ ⓑ $$b^0$$

ⓐ 1 ⓑ 1

14. ⓐ $$13^0$$ ⓑ $$k^0$$

15. ⓐ $$−27^0$$ ⓑ $$−(27^0)$$

ⓐ $$−1$$ ⓑ $$−1$$

16. ⓐ $$−15^0$$ ⓑ $$−(15^0)$$

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

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

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

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

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

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

20. ⓐ $$s^{−8}$$ ⓑ $$10^{−2}$$ ⓒ $$\dfrac{1}{t^{−9}}$$ ⓓ $$\dfrac{1}{10^{−4}}$$

21. ⓐ $$\left(\dfrac{5}{8}\right)^{-2}$$ ⓑ $$\left(−\dfrac{b}{a}\right)^{−2}$$

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

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

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

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

24. ⓐ $$\left(\dfrac{7}{2}\right)^{−3}$$ ⓑ $$\left(−\dfrac{3}{x}\right)^{−3}$$

25. ⓐ $$(−5)^{−2}$$ ⓑ $$−5^{−2}$$ ⓒ $$\left(−\dfrac{1}{5}\right)^{−2}$$ ⓓ $$−\left(\dfrac{1}{5}\right)^{−2}$$

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

26. ⓐ $$−5^{−3}$$ ⓑ $$\left(−\dfrac{1}{5}\right)^{−3}$$ ⓒ $$−\left(\dfrac{1}{5}\right)^{−3}$$ ⓓ $$(−5)^{−3}$$

27. ⓐ $$3·5^{−1}$$ ⓑ $$(3·5)^{−1}$$

ⓐ $$\dfrac{3}{5}$$ ⓑ $$\dfrac{1}{15}$$

28. ⓐ $$3·4^{−2}$$ ⓑ $$(3·4)^{−2}$$

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

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

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

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

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

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

32. ⓐ $$y^{5}·y^{−5}$$ ⓑ $$(pq^{−4})(p^{−6}q^{−3})$$
ⓒ $$(−5m^{4}n^{6})(8m^{−5}n^{−3})$$

33. $$p^{5}·p^{−2}·p^{−4}$$

$$\dfrac{1}{p}$$

34. $$x^{4}·x^{−2}·x^{−3}$$

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

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

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

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

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

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

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

ⓐ $$9x^2y^2$$ ⓑ 1 ⓒ $$\dfrac{1}{25x^4}$$ ⓓ $$\dfrac{16}{y^6}$$

40. ⓐ $$(−4ab)^2$$ ⓑ $$(5x)^0$$ ⓒ $$(4y^3)^{−3}$$ ⓓ $$(−7y^{−3})^2$$

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

ⓐ $$−125a^3b^3$$ ⓑ 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. ⓐ $$(p^2)^5$$ ⓑ $$\left(\dfrac{x}{y}\right)^{−6}$$ ⓒ $$\left(\dfrac{2xy^2}{z}\right)^3$$ ⓓ $$\left(\dfrac{4p^{−3}}{q^2}\right)^2$$

ⓐ $$\dfrac{p^5}{32}$$ ⓑ $$\dfrac{y^6}{x^6}$$ ⓒ $$\dfrac{8x^3y^6}{z^3}$$
ⓓ $$\dfrac{16}{p^6q^4}$$

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. ⓐ $$\left(\dfrac{a}{3b}\right)^4$$ ⓑ $$\left(\dfrac{5}{4m}\right)^{−2}$$ ⓒ $$\left(\dfrac{3a^{−2}b^3}{c^3}\right)^{−2}$$ ⓓ $$\left(\dfrac{p^{−1}q^4}{r^{−4}}\right)^2$$

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

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

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

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

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

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

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

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

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

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

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

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

55. $$(x^2)^4·(x^3)^2$$

$$x^{14}$$

56. $$(y^4)^3·(y^5)^2$$

57. $$(a^2)^6·(a^3)^8$$

$$a^{30}$$

58. $$(b^7)^5·(b^2)^6$$

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

$$2m^{18}$$

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

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

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

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

63. $$(−2a^3b^2)^4$$

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

64. $$(−10u^2v^4)^3$$

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

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

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

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

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

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

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

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

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

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

$$x^{18}y^{18}$$

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

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

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

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

75. ⓐ $$(3x)^2(5x)$$ ⓑ $$(2y)^3(6y)$$

ⓐ $$45x^3$$ ⓑ $$48y^4$$

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

ⓐ $$12r^4$$ ⓑ $$13x^{11}$$

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

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

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

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

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

$$−\dfrac{4000}{n^{12}}$$

Use Scientific Notation

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

82. ⓐ 57,000 ⓑ 0.026

83. ⓐ 340,000 ⓑ 0.041

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

84. ⓐ 8,750,000 ⓑ 0.00000871

85. ⓐ 1,290,000 ⓑ 0.00000103

ⓐ $$1.29\times10^6$$

ⓑ $$103\times10^{−8}$$

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

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

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

ⓐ $$−830$$ ⓑ 0.038

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

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

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

ⓐ 0.02 ⓑ 500,000,000

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

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

ⓐ 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?

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?