7.1 Use Multiplication Properties of Exponents

Izabela Mazur

CHAPTER 7 Powers, Roots, and Scientific Notation

7.1 Use Multiplication Properties of Exponents

Learning Objectives

By the end of this section, you will be able to:

Simplify expressions with exponents
Simplify expressions using the Product Property for Exponents
Simplify expressions using the Power Property for Exponents
Simplify expressions using the Product to a Power Property
Simplify expressions by applying several properties
Multiply monomials

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

We’ll derive the properties of exponents by looking for patterns in several examples.

First, we will look at an example that leads to the Product Property.

What does this mean?
How many factors altogether?

x times x, multiplied by x times x. x times x has two factors. x times x times x has three factors. 2 plus 3 is five factors.

So, we have

Notice that 5 is the sum of the exponents, 2 and 3.

x squared times x cubed is x to the power of 2 plus 3, or x to the fifth power.

We write:

$\begin{array}{c} {x}^{2}\cdot {x}^{3} \\ {x}^{2+3} \\ {x}^{5} \end{array}$

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Product Property for Exponents

If $a$ is a real number, and $m$ and $n$ are counting numbers, then

${a}^{m}\cdot {a}^{n}={a}^{m+n}$

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

$\begin{array}{rcl} {2}^{2}\cdot {2}^{3}& \stackrel{?}{=} & {2}^{2+3} \\ 4\cdot 8 & \stackrel{?}{=} & {2}^{5}\hfill \\ 32& = & 32\checkmark \hfill \end{array}$

EXAMPLE 3

Simplify: ${y}^{5}\cdot {y}^{6}$ .

Solution


Use the product property, a^m \cdot aⁿ = a^m+n.
Simplify.

TRY IT 3.1

Simplify: ${b}^{9}\cdot {b}^{8}$ .

Show answer

${b}^{17}$

TRY IT 3.2

Simplify: ${x}^{12}\cdot {x}^{4}$ .

Show answer

${x}^{16}$

EXAMPLE 4

Simplify: a) ${2}^{5}\cdot {2}^{9}$ b) $3\cdot {3}^{4}$ .

Solution

Use the product property, a^m · aⁿ = a^m+n.

Simplify.
Use the product property, a^m · aⁿ = a^m+n.

Simplify.

TRY IT 4.1

Simplify: a) $5\cdot {5}^{5}$ b) ${4}^{9}\cdot {4}^{9}$ .

Show answer

a) ${5}^{6}$ b) ${4}^{18}$

TRY IT 4.2

Simplify: a) ${7}^{6}\cdot {7}^{8}$ b) $10\cdot {10}^{10}$ .

Show answer

a) ${7}^{14}$ b) ${10}^{11}$

EXAMPLE 5

Simplify: a) ${a}^{7}\cdot a$ b) ${x}^{27}\cdot {x}^{13}$ .

Solution

Rewrite, a = a¹.

Use the product property, a^m · aⁿ = a^m+n.

Simplify.
Notice, the bases are the same, so add the exponents.

Simplify.

TRY IT 5.1

Simplify: a) ${p}^{5}\cdot p$ b) ${y}^{14}\cdot {y}^{29}$ .

Show answer

a) ${p}^{6}$ b) ${y}^{43}$

TRY IT 5.2

Simplify: a) $z\cdot {z}^{7}$ b) ${b}^{15}\cdot {b}^{34}$ .

Show answer

a) ${z}^{8}$ b) ${b}^{49}$

We can extend the Product Property for Exponents to more than two factors.

EXAMPLE 6

Simplify: ${d}^{4}\cdot {d}^{5}\cdot {d}^{2}$ .

Solution


Add the exponents, since bases are the same.
Simplify.

TRY IT 6.1

Simplify: ${x}^{6}\cdot {x}^{4}\cdot {x}^{8}$ .

Show answer

${x}^{18}$

TRY IT 6.2

Simplify: ${b}^{5}\cdot {b}^{9}\cdot {b}^{5}$ .

Show answer

${b}^{19}$

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

1.

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

2.

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

3.

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

4.

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

5.

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

6.

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

7.

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

8.

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

9.

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

10.

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

Simplify Expressions Using the Product Property for Exponents

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

11. ${d}^{3}\cdot {d}^{6}$	12. ${x}^{4}\cdot {x}^{2}$
13. ${n}^{19}\cdot{n}^{12}$	14. ${q}^{27}\cdot {q}^{15}$
15. a) ${4}^{5}\cdot {4}^{9}$ b) ${8}^{9}\cdot 8$	16. a) ${3}^{10}\cdot {3}^{6}$ b) $5\cdot {5}^{4}$
17. a) $y\cdot {y}^{3}$ b) ${z}^{25}\cdot {z}^{8}$	17. a) $y\cdot {y}^{3}$ b) ${z}^{25}\cdot {z}^{8}$
19. $w\cdot {w}^{2}\cdot {w}^{3}$	20. $y\cdot {y}^{3}\cdot {y}^{5}$
21. ${a}^{4}\cdot {a}^{3}\cdot {a}^{9}$	22. ${c}^{5}\cdot {c}^{11}\cdot {c}^{2}$
23. ${m}^{x}\cdot {m}^{3}$	24. ${n}^{y}\cdot {n}^{2}$
25. ${y}^{a}\cdot {y}^{b}$	26. ${x}^{p}\cdot {x}^{q}$

Simplify Expressions Using the Power Property for Exponents

In the following exercises, 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) ${\left({3}^{8}\right)}^{2}$
29. a) ${\left({y}^{3}\right)}^{x}$ b) ${\left({5}^{x}\right)}^{y}$	30. a) ${\left({x}^{2}\right)}^{y}$ b) ${\left({7}^{a}\right)}^{b}$

Simplify Expressions Using the Product to a Power Property

In the following exercises, 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) ${\left(4ab\right)}^{2}$
33. a) ${\left(-4m\right)}^{3}$ b) ${\left(5ab\right)}^{3}$	34. a) ${\left(-7n\right)}^{3}$ b) ${\left(3xyz\right)}^{4}$

Simplify Expressions by Applying Several Properties

In the following exercises, 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) ${\left(2x{y}^{4}\right)}^{5}$

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) ${\left({n}^{3}\right)}^{10}cdot {\left({n}^{5}\right)}^{2}$

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

41.

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

42.

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

43.

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

44.

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

45.

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

Multiply Monomials

In the following exercises, multiply the terms.

46. $\left(6{y}^{7}\right)\left(-3{y}^{4}\right)$	47. $\left(-10{x}^{5}\right)\left(-3{x}^{3}\right)$
48. $\left(-8{u}^{6}\right)\left(-9u\right)$	49. $\left(-6{c}^{4}\right)\left(-12c\right)$
50. $\left(\dfrac{1}{5}{f}^{8}\right)\left(20{f}^{3}\right)$	51. $\left(\dfrac{1}{4}{d}^{5}\right)\left(36{d}^{2}\right)$
52. $\left(4{a}^{3}b\right)\left(9{a}^{2}{b}^{6}\right)$	53. $\left(6{m}^{4}{n}^{3}\right)\left(7m{n}^{5}\right)$
54. $\left(\dfrac{4}{7}r{s}^{2}\right)\left(14r{s}^{3}\right)$	55. $\left(\dfrac{5}{8}{x}^{3}y\right)\left(24{x}^{5}y\right)$
56. $\left(\dfrac{2}{3}{x}^{2}y\right)\left(\dfrac{3}{4}x{y}^{2}\right)$	56. $\left(\dfrac{2}{3}{x}^{2}y\right)\left(\dfrac{3}{4}x{y}^{2}\right)$

Mixed Practice

In the following exercises, simplify each expression.

58. ${\left({x}^{2}\right)}^{4}\cdot {\left({x}^{3}\right)}^{2}$	59. ${\left({y}^{4}\right)}^{3}\cdot {\left({y}^{5}\right)}^{2}$
60. ${\left({a}^{2}\right)}^{6}\cdot{\left({a}^{3}\right)}^{8}$	61. ${\left({b}^{7}\right)}^{5}\cdot {\left({b}^{2}\right)}^{6}$
62. ${\left(2{m}^{6}\right)}^{3}$	63. ${\left(3{y}^{2}\right)}^{4}$
64. ${\left(10{x}^{2}y\right)}^{3}$	65. ${\left(2m{n}^{4}\right)}^{5}$
66. ${\left(-2{a}^{3}{b}^{2}\right)}^{4}$	67. ${\left(-10{u}^{2}{v}^{4}\right)}^{3}$
68. ${\left(\dfrac{2}{3}{x}^{2}y\right)}^{3}$	69. ${\left(\dfrac{7}{9}p{q}^{4}\right)}^{2}$
70. ${\left(8{a}^{3}\right)}^{2}{\left(2a\right)}^{4}$	71. ${\left(5{r}^{2}\right)}^{3}{\left(3r\right)}^{2}$
72. ${\left(10{p}^{4}\right)}^{3}{\left(5{p}^{6}\right)}^{2}$	73. ${\left(4{x}^{3}\right)}^{3}{\left(2{x}^{5}\right)}^{4}$
74. ${\left(\dfrac{1}{2}{x}^{2}{y}^{3}\right)}^{4}{\left(4{x}^{5}{y}^{3}\right)}^{2}$	75. ${\left(\dfrac{1}{3}{m}^{3}{n}^{2}\right)}^{4}{\left(9{m}^{8}{n}^{3}\right)}^{2}$
76. ${\left(3{m}^{2}n\right)}^{2}{\left(2m{n}^{5}\right)}^{4}$	77. ${\left(2p{q}^{4}\right)}^{3}{\left(5{p}^{6}q\right)}^{2}$

Everyday Math

78. 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.

Round	Number of people
1	10
2	${10}^{2}$
3	${10}^{3}$
…	…
6	?

79. 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 $35,000, his salary after 1 year was $\text{\$}35,000\left(1.03\right)$ , after 2 years was $\text{\$}35,000{\left(1.03\right)}^{2}$ , after 3 years was $\text{\$}35,000{\left(1.03\right)}^{3}$ , 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.

Year	Salary
1	$\text{\$}35,000\left(1.03\right)$
2	$\text{\$}35,000{\left(1.03\right)}^{2}$
3	$\text{\$}35,000{\left(1.03\right)}^{3}$
…	…
10	?

80. 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 $1,000, the cost for the first week would be $\text{\$}1,000\left(0.70\right)$ and the cost of the item during the second week would be $\text{\$}1,000{\left(0.70\right)}^{2}$ . 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.

Week	Cost
1	$\text{\$}1,000\left(0.70\right)$
2	$\text{\$}1,000{\left(0.70\right)}^{2}$
3
…	…
5	?

81. 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 ?20,000, the value at the end of the first year would be $\text{\$}20,000\left(0.90\right)$ and the value of the car after the end of the second year would be $\text{\$}20,000{\left(0.90\right)}^{2}$ . 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.

Week	Cost
1	$\text{\$}20,000\left(0.90\right)$
2	$\text{\$}20,000{\left(0.90\right)}^{2}$
3
4	…
8	?

Writing Exercises

82. Use the Product Property for Exponents to explain why $x\cdot x={x}^{2}$ .	83. Explain why $-{5}^{3}={\left(-5\right)}^{3}$ but $-{5}^{4}\ne {\left(-5\right)}^{4}$ .
84. Jorge thinks ${\left(\dfrac{1}{2}\right)}^{2}$ is 1. What is wrong with his reasoning?	85. Explain why ${x}^{3}\cdot {x}^{5}$ is ${x}^{8}$ , and not ${x}^{15}$

License

Icon for the Creative Commons Attribution 4.0 International License

a)	${4}^{3}$
Multiply three factors of 4.	4 · 4 · 4
Simplify.	$64$
b)	${7}^{1}$
Multiply one factor of 7.	$7$
c)	${\left(\dfrac{5}{6}\right)}^{2}$
Multiply two factors.	$\left(\dfrac{5}{6}\right)\left(\dfrac{5}{6}\right)$
Simplify.	$\dfrac{25}{36}$
d)	${\left(0.63\right)}^{2}$
Multiply two factors.	$\left(0.63\right)\left(0.63\right)$
Simplify.	$0.3969$

a)	${\left(-5\right)}^{4}$
Multiply four factors of $-5$ .	$\left(-5\right)\left(-5\right)\left(-5\right)\left(-5\right)$
Simplify.	$625$
b)	$-{5}^{4}$
Multiply four factors of 5.	-(5 · 5 · 5 · 5)
Simplify.	$-625$

Product Property	${a}^{m}\cdot {a}^{n} ={a}^{m+n}$
Power Property	$({a}^{m})^{n} = {a}^{m \cdot n}$
Product to a Power	$(ab)^{m} = {a}^{m}{b}^{m}$

a)	${\left({y}^{3}\right)}^{6}{\left({y}^{5}\right)}^{4}$
Use the Power Property.	${y}^{18}\cdot {y}^{20}$
Add the exponents.	${y}^{38}$
b)	${\left(-6{x}^{4}{y}^{5}\right)}^{2}$
Use the Product to a Power Property.	${\left(-6\right)}^{2}{\left({x}^{4}\right)}^{2}{\left({y}^{5}\right)}^{2}$
Use the Power Property.	${\left(-6\right)}^{2}$
Simplify.	$36{x}^{8}{y}^{10}$

a)	${\left(5m\right)}^{2}\left(3{m}^{3}\right)$
Raise $5m$ to the second power.	${5}^{2}{m}^{2}\cdot 3{m}^{3}$
Simplify.	$25{m}^{2}\cdot 3{m}^{3}$
Use the Commutative Property.	$25\cdot 3\cdot {m}^{2}\cdot {m}^{3}$
Multiply the constants and add the exponents.	$75{m}^{5}$
b)	${\left(3{x}^{2}y\right)}^{4}{\left(2x{y}^{2}\right)}^{3}$
Use the Product to a Power Property.	$\left({3}^{4}{x}^{8}{y}^{4}\right)\left({2}^{3}{x}^{3}{y}^{6}\right)$
Simplify.	$\left(81{x}^{8}{y}^{4}\right)\left(8{x}^{3}{y}^{6}\right)$
Use the Commutative Property.	$81\cdot 8\cdot {x}^{8}\cdot {x}^{3}\cdot {y}^{4}\cdot {y}^{6}$
Multiply the constants and add the exponents.	$648{x}^{11}{y}^{10}$


Rewrite, a = a¹.
Use the product property, a^m · aⁿ = a^m+n.
Simplify.


Use Power of a Product Property, (ab)^m = a^mb^m.
Simplify.

	$\left(3{x}^{2}\right)\left(-4{x}^{3}\right)$
Use the Commutative Property to rearrange the terms.	$3\cdot \left(-4\right)\cdot {x}^{2}\cdot {x}^{3}$
Multiply.	$-12{x}^{5}$

	$\left(\dfrac{5}{6}{x}^{3}y\right)\left(12x{y}^{2}\right)$
Use the Commutative Property to rearrange the terms.	$\dfrac{5}{6}\cdot 12\cdot {x}^{3}\cdot x \cdot y \cdot {y}^{2}$
Multiply.	$10{x}^{4}{y}^{3}$

2. a) 10,000 b) 17 c) $\dfrac{4}{81}$ d) 0.125	4. a) 512 b) 8 c) $\dfrac{27}{64}$ d) 0.064
6. a) 64 b) $-64$	8. a) $-\dfrac{4}{9}$ b) $\dfrac{4}{9}$
10. a) $-0.0001$ b) $0.0001$	12. ${x}^{6}$
14. ${q}^{42}$	16. a) ${3}^{16}$ b) ${5}^{5}$
18. a) ${w}^{6}$ b) ${u}^{94}$	20. ${y}^{9}$
22. ${c}^{18}$	24. ${n}^{y+2}$
26. ${x}^{p+q}$	28. a) ${b}^{14}$ b) ${3}^{16}$
30. a) ${x}^{2y}$ b) ${7}^{ab}$	32. a) $25{x}^{2}$ b) $16{a}^{2}{b}^{2}$
34. a) $-343{n}^{3}$ b) $81{x}^{4}{y}^{4}{z}^{4}$	36. a) ${w}^{22}$ b) $32{x}^{5}{y}^{20}$
38. a) $-1000{q}^{6}{p}^{12}$ b) ${n}^{40}$	40. a) $48{y}^{4}$ b) $25,000{k}^{24}$
42. a) $432{b}^{5}$ b) $\dfrac{1}{200}{j}^{16}$	44. a) $128{r}^{8}$ b) $\dfrac{1}{200}{j}^{16}$
46. $-18{y}^{11}$	48. $72{u}^{7}$
50. $4{f}^{11}$	52. $36{a}^{5}{b}^{7}$
54. $8{r}^{2}{s}^{5}$	56. $\dfrac{1}{2}{x}^{3}{y}^{3}$
58. ${x}^{14}$	60. ${a}^{36}$
62. $8{m}^{18}$	64. $1000{x}^{6}{y}^{3}$
66. $16{a}^{12}{b}^{8}$	68. $\dfrac{8}{27}{x}^{6}{y}^{3}$
70. $1024{a}^{10}$	72. $25000{p}^{24}$
74. ${x}^{18}{y}^{18}$	76. $144{m}^{8}{n}^{22}$
78. $1,000,000$	80. $168.07
82. Answers will vary.	84. Answers will vary.


Use the power property, (a^m)ⁿ = a^{m · n}.
Simplify.

We write:	${\left(2x\right)}^{3}$
	${2}^{3}\cdot {x}^{3}$

Simplify Expressions with Exponents

Simplify Expressions Using the Product Property for Exponents

Simplify Expressions Using the Power Property for Exponents

Simplify Expressions Using the Product to a Power Property

Simplify Expressions by Applying Several Properties

Multiply Monomials

Key Concepts

Practice Makes Perfect

Simplify Expressions with Exponents

Simplify Expressions Using the Product Property for Exponents

Simplify Expressions Using the Power Property for Exponents

Simplify Expressions Using the Product to a Power Property

Simplify Expressions by Applying Several Properties

Multiply Monomials

Mixed Practice

Everyday Math

Writing Exercises

Answers

Attributions

License

Share This Book