# Exercises: Use Multiplication Properties of Exponents (5.2)

## Exercises: Simplify Expressions with Exponents

Instructions: For questions 1-10, simplify each expression with exponents.

1.

a. ${3}^{5}$
b. ${9}^{1}$
c. ${\left(\frac{1}{3}\right)}^{2}$
d. ${\left(0.2\right)}^{4}$

2.

a. ${10}^{4}$
b. ${17}^{1}$
c. ${\left(\frac{2}{9}\right)}^{2}$
d. ${\left(0.5\right)}^{3}$

Solution

a. $10\text{,}000$
b. $17$
c. $\frac{4}{81}$
d. $0.125$

3.

a. ${2}^{6}$
b. ${14}^{1}$
c. ${\left(\frac{2}{5}\right)}^{3}$
d. ${\left(0.7\right)}^{2}$

4.

a. ${8}^{3}$
b. ${8}^{1}$
c. ${\left(\frac{3}{4}\right)}^{3}$
d. ${\left(0.4\right)}^{3}$

Solution

a. $512$
b. $8$
c. $\frac{27}{64}$
d. $0.064$

5.

a. ${\left(-6\right)}^{4}$
b. $-{6}^{4}$

6.

a. ${\left(-2\right)}^{6}$
b. $-{2}^{6}$

Solution

a. $64$
b. $-64$

7.

a. $-{\left(\frac{1}{4}\right)}^{4}$
b. ${\left(-\frac{1}{4}\right)}^{4}$

8.

a. $-{\left(\frac{2}{3}\right)}^{2}$
b. ${\left(-\frac{2}{3}\right)}^{2}$

Solution

a. $-\frac{4}{9}$
b. $\frac{4}{9}$

9.

a.$-{0.5}^{2}$
b.${\left(-0.5\right)}^{2}$

10.

a. $-{0.1}^{4}$
b. ${\left(-0.1\right)}^{4}$

Solution

a. $-0.0001$
b. $0.0001$

## Exercises: Simplify Expressions Using the Product Property for Exponents

Instructions: For questions 11-26, simplify each expression using the Product Property for Exponents.

11. ${d}^{3}\cdot {d}^{6}$

12. ${x}^{4}\cdot {x}^{2}$
Solution

${x}^{6}$

13. ${n}^{19}\cdot {n}^{12}$

14. ${q}^{27}\cdot {q}^{15}$
Solution

${q}^{42}$

15.

a. ${4}^{5}\cdot {4}^{9}$
b. ${8}^{9}\cdot 8$

16.

a. ${3}^{10}\cdot {3}^{6}$
b.$5\cdot {5}^{4}$

Solution

a. ${3}^{16}$
b. ${5}^{5}$

17.

a. $y\cdot {y}^{3}$
b. ${z}^{25}\cdot {z}^{8}$

18.

a. ${w}^{5}\cdot w$
b. ${u}^{41}\cdot {u}^{53}$

Solution

a. ${w}^{6}$
b. ${u}^{94}$

19. $w\cdot {w}^{2}\cdot {w}^{3}$

20. $y\cdot {y}^{3}\cdot {y}^{5}$
Solution

${y}^{9}$

21. ${a}^{4}\cdot {a}^{3}\cdot {a}^{9}$

22. ${c}^{5}\cdot {c}^{11}\cdot {c}^{2}$
Solution

${c}^{18}$

23. ${m}^{x}\cdot {m}^{3}$

24. ${n}^{y}\cdot {n}^{2}$
Solution

${n}^{y+2}$

25. ${y}^{a}\cdot {y}^{b}$

26. ${x}^{p}\cdot {x}^{q}$
Solution

${x}^{p+q}$

## Exercises: Simplify Expressions Using the Power Property for Exponents

Instructions: For questions 27-30, simplify each expression using the Power Property for Exponents.

27.

a. ${\left({m}^{4}\right)}^{2}$
b. ${\left({10}^{3}\right)}^{6}$

28.

a. ${\left({b}^{2}\right)}^{7}$
b. ${\left({3}^{8}\right)}^{2}$

Solution

a. ${b}^{14}$
b. ${3}^{16}$

29.

a. ${\left({y}^{3}\right)}^{x}$
b. ${\left({5}^{x}\right)}^{y}$

30.

a. ${\left({x}^{2}\right)}^{y}$
b. ${\left({7}^{a}\right)}^{b}$

Solution

a. ${x}^{2y}$
b. ${7}^{ab}$

## Exercises: Simplify Expressions Using the Product to a Power Property

Instructions: For questions 31-34, simplify each expression using the Product to a Power Property.

31.

a. ${\left(6a\right)}^{2}$
b. ${\left(3xy\right)}^{2}$

32.

a. ${\left(5x\right)}^{2}$
b. ${\left(4ab\right)}^{2}$

Solution

a. $25{x}^{2}$
b. $16{a}^{2}{b}^{2}$

33.

a. ${\left(-4m\right)}^{3}$
b. ${\left(5ab\right)}^{3}$

34.

a. ${\left(-7n\right)}^{3}$
b. ${\left(3xyz\right)}^{4}$

Solution

a.$-343{n}^{3}$b.$81{x}^{4}{y}^{4}{z}^{4}$

## Exercises: Simplify Expressions by Applying Several Properties

Instructions: For questions 35-45, simplify each expression.

35.

a. ${\left({y}^{2}\right)}^{4}\cdot {\left({y}^{3}\right)}^{2}$
b. ${\left(10{a}^{2}b\right)}^{3}$

36.

a. ${\left({w}^{4}\right)}^{3}\cdot {\left({w}^{5}\right)}^{2}$
b. ${\left(2x{y}^{4}\right)}^{5}$

Solution

a. ${w}^{22}$
b. $32{x}^{5}{y}^{20}$

37.

a. ${\left(-2{r}^{3}{s}^{2}\right)}^{4}$
b. ${\left({m}^{5}\right)}^{3}\cdot {\left({m}^{9}\right)}^{4}$

38.

a. ${\left(-10{q}^{2}{p}^{4}\right)}^{3}$
b. ${\left({n}^{3}\right)}^{10}\cdot {\left({n}^{5}\right)}^{2}$

Solution

a. $-1000{q}^{6}{p}^{12}$
b. ${n}^{40}$

39.

a. ${\left(3x\right)}^{2}\left(5x\right)$
b. ${\left(5{t}^{2}\right)}^{3}{\left(3t\right)}^{2}$

40.

a. ${\left(2y\right)}^{3}\left(6y\right)$
b. ${\left(10{k}^{4}\right)}^{3}{\left(5{k}^{6}\right)}^{2}$

Solution

a. $48{y}^{4}$
b. $25\text{,}000{k}^{24}$

41.

a. ${\left(5a\right)}^{2}{\left(2a\right)}^{3}$
b. ${\left(\frac{1}{2}{y}^{2}\right)}^{3}{\left(\frac{2}{3}y\right)}^{2}$

42.

a. ${\left(4b\right)}^{2}{\left(3b\right)}^{3}$
b. ${\left(\frac{1}{2}{j}^{2}\right)}^{5}{\left(\frac{2}{5}{j}^{3}\right)}^{2}$

Solution

a. $432{b}^{5}$
b. $\frac{1}{200}{j}^{16}$

43.

a. ${\left(\frac{2}{5}{x}^{2}y\right)}^{3}$
b. ${\left(\frac{8}{9}x{y}^{4}\right)}^{2}$

44.

a. ${\left(2{r}^{2}\right)}^{3}{\left(4r\right)}^{2}$
b. ${\left(3{x}^{3}\right)}^{3}{\left({x}^{5}\right)}^{4}$

Solution

a. $128{r}^{8}$
b. $\frac{1}{200}{j}^{16}$

45.

a. ${\left({m}^{2}n\right)}^{2}{\left(2m{n}^{5}\right)}^{4}$
b. ${\left(3p{q}^{4}\right)}^{2}{\left(6{p}^{6}q\right)}^{2}$

## Exercises: Multiply Monomials

Instructions: For questions 46-57, multiply the monomials.

46. $\left(6{y}^{7}\right)\left(-3{y}^{4}\right)$
Solution

$-18{y}^{11}$

47. $\left(-10{x}^{5}\right)\left(-3{x}^{3}\right)$

48. $\left(-8{u}^{6}\right)\left(-9u\right)$
Solution

$72{u}^{7}$

49. $\left(-6{c}^{4}\right)\left(-12c\right)$

50. $\left(\frac{1}{5}{f}^{8}\right)\left(20{f}^{3}\right)$
Solution

$4{f}^{11}$

51. $\left(\frac{1}{4}{d}^{5}\right)\left(36{d}^{2}\right)$

52. $\left(4{a}^{3}b\right)\left(9{a}^{2}{b}^{6}\right)$
Solution

$36{a}^{5}{b}^{7}$

53. $\left(6{m}^{4}{n}^{3}\right)\left(7m{n}^{5}\right)$

54. $\left(\frac{4}{7}r{s}^{2}\right)\left(14r{s}^{3}\right)$
Solution

$8{r}^{2}{s}^{5}$

55. $\left(\frac{5}{8}{x}^{3}y\right)\left(24{x}^{5}y\right)$

56. $\left(\frac{2}{3}{x}^{2}y\right)\left(\frac{3}{4}x{y}^{2}\right)$
Solution

$\frac{1}{2}{x}^{3}{y}^{3}$

57. $\left(\frac{3}{5}{m}^{3}{n}^{2}\right)\left(\frac{5}{9}{m}^{2}{n}^{3}\right)$

## Exercises: Mixed Practice

Instructions: For questions 58-77, simplify each expression.

58. ${\left({x}^{2}\right)}^{4}\cdot {\left({x}^{3}\right)}^{2}$
Solution

${x}^{14}$

59. ${\left({y}^{4}\right)}^{3}\cdot {\left({y}^{5}\right)}^{2}$

60. ${\left({a}^{2}\right)}^{6}\cdot {\left({a}^{3}\right)}^{8}$
Solution

${a}^{36}$

61. ${\left({b}^{7}\right)}^{5}\cdot {\left({b}^{2}\right)}^{6}$

62. ${\left(2{m}^{6}\right)}^{3}$
Solution

$8{m}^{18}$

63. ${\left(3{y}^{2}\right)}^{4}$

64. ${\left(10{x}^{2}y\right)}^{3}$
Solution

$1000{x}^{6}{y}^{3}$

65. ${\left(2m{n}^{4}\right)}^{5}$

66. ${\left(-2{a}^{3}{b}^{2}\right)}^{4}$
Solution

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

67. ${\left(-10{u}^{2}{v}^{4}\right)}^{3}$

68. ${\left(\frac{2}{3}{x}^{2}y\right)}^{3}$
Solution

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

69. ${\left(\frac{7}{9}p{q}^{4}\right)}^{2}$

70. ${\left(8{a}^{3}\right)}^{2}{\left(2a\right)}^{4}$
Solution

$1024{a}^{10}$

71. ${\left(5{r}^{2}\right)}^{3}{\left(3r\right)}^{2}$

72. ${\left(10{p}^{4}\right)}^{3}{\left(5{p}^{6}\right)}^{2}$
Solution

$25000{p}^{24}$

73. ${\left(4{x}^{3}\right)}^{3}{\left(2{x}^{5}\right)}^{4}$

74. ${\left(\frac{1}{2}{x}^{2}{y}^{3}\right)}^{4}{\left(4{x}^{5}{y}^{3}\right)}^{2}$
Solution

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

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

76. ${\left(3{m}^{2}n\right)}^{2}{\left(2m{n}^{5}\right)}^{4}$
Solution

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

77. ${\left(2p{q}^{4}\right)}^{3}{\left(5{p}^{6}q\right)}^{2}$

## Exercises: Use the Definition of a Negative Exponent

Instructions: For questions 78-105, simplify.

78.

a. ${4}^{-2}$
b. ${10}^{-3}$

79.

a. ${3}^{-4}$
b. ${10}^{-2}$

Solution

a. $\frac{1}{81}$
b. $\frac{1}{100}$

80.

a. ${5}^{-3}$
b. ${10}^{-5}$

81.

a. ${2}^{-8}$
b. ${10}^{-2}$

Solution

a. $\frac{1}{256}$
b. $\frac{1}{100}$

82.

a. $\frac{1}{{c}^{-5}}$
b. $\frac{1}{{3}^{-2}}$

83.

a. $\frac{1}{{c}^{-5}}$
b. $\frac{1}{{5}^{-2}}$

Solution

a. ${c}^{5}$
b. $25$

84.

a. $\frac{1}{{q}^{-10}}$
b. $\frac{1}{{10}^{-3}}$

85.

a. $\frac{1}{{t}^{-9}}$
b. $\frac{1}{{10}^{-4}}$

Solution

a. ${t}^{9}$
b. $10000$

86.

a. ${\left(\frac{5}{8}\right)}^{-2}$
b. ${\left(-\frac{3m}{n}\right)}^{-2}$

87.

a. ${\left(\frac{3}{10}\right)}^{-2}$
b. ${\left(-\frac{2}{cd}\right)}^{-3}$

Solution

a. $\frac{100}{9}$
b. $-\frac{{c}^{3}{d}^{3}}{8}$

88.

a. ${\left(\frac{4}{9}\right)}^{-3}$
b. ${\left(-\frac{{u}^{2}}{2v}\right)}^{-5}$

89.

a. ${\left(\frac{7}{2}\right)}^{-3}$
b. ${\left(-\frac{3}{x{y}^{2}}\right)}^{-3}$

Solution

a. $\frac{8}{343}$
b. $-\frac{{x}^{3}{y}^{6}}{27}$

90.

a. ${\left(-5\right)}^{-2}$
b. $-{5}^{-2}$
c. ${\left(-\frac{1}{5}\right)}^{-2}$
d. $-{\left(\frac{1}{5}\right)}^{-2}$

91.

a. ${\left(-7\right)}^{-2}$
b. $-{7}^{-2}$
c. ${\left(-\frac{1}{7}\right)}^{-2}$
d. $-{\left(\frac{1}{7}\right)}^{-2}$

Solution

a. $\frac{1}{49}$
b. $-\frac{1}{49}$
c. $49$
d. $-49$

92.

a. $-{3}^{-3}$
b. ${\left(-\frac{1}{3}\right)}^{-3}$
c. $-{\left(\frac{1}{3}\right)}^{-3}$
d. ${\left(-3\right)}^{-3}$

93.

a. $-{5}^{-3}$
b. ${\left(-\frac{1}{5}\right)}^{-3}$
c. $-{\left(\frac{1}{5}\right)}^{-3}$
d. ${\left(-5\right)}^{-3}$

Solution

a. $-\frac{1}{125}$
b. $-125$
c. $-125$
d. $-\frac{1}{125}$

94.

a. $3\cdot {5}^{-1}$
b. ${\left(3\cdot 5\right)}^{-1}$

95.

a. $2\cdot {5}^{-1}$
b. ${\left(2\cdot 5\right)}^{-1}$

Solution

a. $\frac{2}{5}$
b. $\frac{1}{10}$

96.

a. $4\cdot {5}^{-2}$
b. ${\left(4\cdot 5\right)}^{-2}$

97.

a. $3\cdot {4}^{-2}$
b. ${\left(3\cdot 4\right)}^{-2}$

Solution

a. $\frac{3}{16}$
b. $\frac{1}{144}$

98.

a. ${m}^{-4}$
b. ${\left({x}^{3}\right)}^{-4}$

99.

a. ${b}^{-5}$
b. ${\left({k}^{2}\right)}^{-5}$

Solution

a. $\frac{1}{{b}^{5}}$
b. $\frac{1}{{k}^{10}}$

100.

a. ${p}^{-10}$
b. ${\left({q}^{6}\right)}^{-8}$

101.

a. ${s}^{-8}$
b. ${\left({a}^{9}\right)}^{-10}$

Solution

a. $\frac{1}{{s}^{8}}$
b. $\frac{1}{{a}^{90}}$

102.

a. $7{n}^{-1}$
b. ${\left(7n\right)}^{-1}$
c. ${\left(-7n\right)}^{-1}$

103.

a. $6{r}^{-1}$
b. ${\left(6r\right)}^{-1}$
c. ${\left(-6r\right)}^{-1}$

Solution

a. $\frac{6}{r}$
b. $\frac{1}{6r}$
c. $-\frac{1}{6r}$

104.

a. ${\left(3p\right)}^{-2}$
b. $3{p}^{-2}$
c. $-3{p}^{-2}$

105.

a. ${\left(2q\right)}^{-4}$
b. $2{q}^{-4}$
c. $-2{q}^{-4}$

Solution

a. $\frac{1}{16{q}^{4}}$
b. $\frac{2}{{q}^{4}}$
c. $-\frac{2}{{q}^{4}}$

## Exercises: Simplify Expressions with Integer Exponents

Instructions: For questions 106-127, simplify.

106.

a. ${b}^{4}{b}^{-8}$
b. ${r}^{-2}{r}^{5}$
c. ${x}^{-7}{x}^{-3}$

107.

a. ${s}^{3}\cdot {s}^{-7}$
b. ${q}^{-8}\cdot {q}^{3}$
c. ${y}^{-2}\cdot {y}^{-5}$

Solution

a. $\frac{1}{{s}^{4}}$
b. $\frac{1}{{q}^{5}}$
c. $\frac{1}{{y}^{7}}$

108.

a. ${a}^{3}\cdot {a}^{-3}$
b. $a\cdot {a}^{3}$
c. $a\cdot {a}^{-3}$

109.

a. ${y}^{5}\cdot {y}^{-5}$
b. $y\cdot {y}^{5}$
c. $y\cdot {y}^{-5}$

Solution

a. $1$
b. ${y}^{6}$
c. $\frac{1}{{y}^{4}}$

110. ${p}^{5}\cdot {p}^{-2}\cdot {p}^{-4}$

111. ${x}^{4}\cdot {x}^{-2}\cdot {x}^{-3}$
Solution

$\frac{1}{x}$

112. $\left({w}^{4}{x}^{-5}\right)\left({w}^{-2}{x}^{-4}\right)$

113. $\left({m}^{3}{n}^{-3}\right)\left({m}^{-5}{n}^{-1}\right)$
Solution

$\frac{1}{{m}^{2}{n}^{4}}$

114. $\left(u{v}^{-2}\right)\left({u}^{-5}{v}^{-3}\right)$

115. $\left(p{q}^{-4}\right)\left({p}^{-6}{q}^{-3}\right)$
Solution

$\frac{1}{{p}^{5}{q}^{7}}$

116. $\left(-6{c}^{-3}{d}^{9}\right)\left(2{c}^{4}{d}^{-5}\right)$

117. $\left(-2{j}^{-5}{k}^{8}\right)\left(7{j}^{2}{k}^{-3}\right)$
Solution

$-\frac{14{k}^{5}}{{j}^{3}}$

118. $\left(-4{r}^{-2}{s}^{-8}\right)\left(9{r}^{4}{s}^{3}\right)$

119. $\left(-5{m}^{4}{n}^{6}\right)\left(8{m}^{-5}{n}^{-3}\right)$
Solution

$-\frac{40{n}^{3}}{m}$

120. ${\left(5{x}^{2}\right)}^{-2}$

121. ${\left(4{y}^{3}\right)}^{-3}$
Solution

$\frac{1}{64{y}^{9}}$

122. ${\left(3{z}^{-3}\right)}^{2}$

123. ${\left(2{p}^{-5}\right)}^{2}$
Solution

$\frac{4}{{p}^{10}}$

124. $\frac{{t}^{9}}{{t}^{-3}}$

125. $\frac{{n}^{5}}{{n}^{-2}}$
Solution

${n}^{7}$

126. $\frac{{x}^{-7}}{{x}^{-3}}$

127. $\frac{{y}^{-5}}{{y}^{-10}}$
Solution

${y}^{5}$

## Exercises: Everyday Math

Instructions: For questions 128-131, answer the given everyday math word problems.

128. Email. Kate emails a flyer to ten of her friends and tells them to forward it to ten of their friends, who forward it to ten of their friends, and so on. The number of people who receive the email on the second round is ${10}^{2}$, on the third round is ${10}^{3}$, as shown in the table below. How many people will receive the email on the sixth round? Simplify the expression to show the number of people who receive the email.

Table 5P.2.1: Number of People Receving Email per Round
Round Number of people
1 $10$
2 ${10}^{2}$
3 ${10}^{3}$
6 ?

Solution

$1\text{,}000\text{,}000$

129. Salary. Jamal’s boss gives him a $3\%$ raise every year on his birthday. This means that each year, Jamal’s salary is $1.03$ times his last year’s salary. If his original salary was $35\text{,}000$, his salary after $1$ year was $35\text{,}000\left(1.03\right)$, after $2$ years was $35\text{,}000{\left(1.03\right)}^{2}$, after $3$ years was $35\text{,}000{\left(1.03\right)}^{3}$, as shown in the table below. What will Jamal’s salary be after $10$ years? Simplify the expression, to show Jamal’s salary in dollars.

Figure 5P.2.2: Jamal’s Salary by Year
Year Salary
1 $35\text{,}000\left(1.03\right)$
2 $35\text{,}000{\left(1.03\right)}^{2}$
3 $35\text{,}000{\left(1.03\right)}^{3}$
10 ?

130. Clearance. A department store is clearing out merchandise in order to make room for new inventory. The plan is to mark down items by $30\%$ each week. This means that each week the cost of an item is $70\%$ of the previous week’s cost. If the original cost of a sofa was $1\text{,}000$, the cost for the first week would be $1\text{,}000\left(0.70\right)$ and the cost of the item during the second week would be $1\text{,}000{\left(0.70\right)}^{2}$. Complete the table shown below. What will be the cost of the sofa during the fifth week? Simplify the expression, to show the cost in dollars.

Figure 5P.2.3: Cost of Merchandise per Week
Week Cost
1 $1\text{,}000\left(0.70\right)$
2 $1\text{,}000{\left(0.70\right)}^{2}$
3
5 ?

Solution

$168.07$

131. Depreciation. Once a new car is driven away from the dealer, it begins to lose value. Each year, a car loses $10\%$ of its value. This means that each year the value of a car is $90\%$ of the previous year’s value. If a new car was purchased for $20\text{,}000$, the value at the end of the first year would be $20\text{,}000\left(0.90\right)$ and the value of the car after the end of the second year would be $20\text{,}000{\left(0.90\right)}^{2}$. Complete the table shown below. What will be the value of the car at the end of the eighth year? Simplify the expression, to show the value in dollars.

Figure 5P.2.4: Depreciation of a New Car over 8 years
Year Cost
1 $20\text{,}000\left(0.90\right)$
2 $20\text{,}000{\left(0.90\right)}^{2}$
3
8 ?

## Exercises: Writing Exercises

Instructions: For questions 132-135,
132. Use the Product Property for Exponents to explain why $x\cdot x={x}^{2}$.
Solution

133. Explain why $-{5}^{3}={\left(-5\right)}^{3}$ but $-{5}^{4}\ne {\left(-5\right)}^{4}$.

134. Jorge thinks ${\left(\frac{1}{2}\right)}^{2}$ is $1$. What is wrong with his reasoning?
Solution

135. Explain why ${x}^{3}\cdot {x}^{5}$ is ${x}^{8}$, and not ${x}^{15}$.
a. Explain the meaning of the exponent in the expression ${2}^{3}$.
b. Explain the meaning of the exponent in the expression ${2}^{-3}$.