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Elementary Algebra 2e

6.2 Use Multiplication Properties of Exponents

Elementary Algebra 2e6.2 Use Multiplication Properties of Exponents
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

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
Be Prepared 6.4

Before you get started, take this readiness quiz.

Simplify: 34·34.34·34.
If you missed this problem, review Example 1.68.

Be Prepared 6.5

Simplify: (−2)(−2)(−2).(−2)(−2)(−2).
If you missed this problem, review Example 1.50.

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, 2424 means to multiply 2 by itself 4 times, so 2424 means 2·2·2·22·2·2·2.

Let’s review the vocabulary for expressions with exponents.

Exponential Notation

This figure has two columns. In the left column is a to the m power. The m is labeled in blue as an exponent. The a is labeled in red as the base. In the right column is the text “a to the m power means multiply m factors of a.” Below this is a to the m power equals a times a times a times a, followed by an ellipsis, with “m factors” written below in blue.

This is read aa to the mthmth power.

In the expression amam, the exponent mm tells us how many times we use the base aa as a factor.

This figure has two columns. The left column contains 4 cubed. Below this is 4 times 4 times 4, with “3 factors” written below in blue. The right column contains negative 9 to the fifth power. Below this is negative 9 times negative 9 times negative 9 times negative 9 times negative 9, with “5 factors” written below in blue.

Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.

Example 6.16

Simplify: 4343 7171 (56)2(56)2 (0.63)2.(0.63)2.

Try It 6.31

Simplify: 6363 151151 (37)2(37)2 (0.43)2.(0.43)2.

Try It 6.32

Simplify: 2525 211211 (25)3(25)3 (0.218)2.(0.218)2.

Example 6.17

Simplify: (−5)4(−5)4 54.54.

Try It 6.33

Simplify: (−3)4(−3)4 34.34.

Try It 6.34

Simplify: (−13)2(−13)2 132.132.

Notice the similarities and differences in Example 6.17 and Example 6.17! Why are the answers different? As we follow the order of operations in part the parentheses tell us to raise the (−5)(−5) to the 4th power. In part we raise just the 5 to the 4th power and then take the opposite.


Simplify Expressions Using the Product Property for Exponents

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.

x squared times x cubed.
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 x to the fifth power.
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:

x2·x3x2+3x5x2·x3x2+3x5

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

Product Property for Exponents

If aa is a real number, and mandnmandn are counting numbers, then

am·an=am+nam·an=am+n

To multiply with like bases, add the exponents.

An example with numbers helps to verify this property.

22·23=?22+34·8=?2532=3222·23=?22+34·8=?2532=32

Example 6.18

Simplify: y5·y6.y5·y6.

Try It 6.35

Simplify: b9·b8.b9·b8.

Try It 6.36

Simplify: x12·x4.x12·x4.

Example 6.19

Simplify: 25·2925·29 3·34.3·34.

Try It 6.37

Simplify: 5·555·55 49·49.49·49.

Try It 6.38

Simplify: 76·7876·78 10·1010.10·1010.

Example 6.20

Simplify: a7·aa7·a x27·x13.x27·x13.

Try It 6.39

Simplify: p5·pp5·p y14·y29.y14·y29.

Try It 6.40

Simplify: z·z7z·z7 b15·b34.b15·b34.

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

Example 6.21

Simplify: d4·d5·d2.d4·d5·d2.

Try It 6.41

Simplify: x6·x4·x8.x6·x4·x8.

Try It 6.42

Simplify: b5·b9·b5.b5·b9·b5.

Simplify Expressions Using the Power Property for Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

x squared, in parentheses, cubed.
What does this mean?
How many factors altogether?
x squared cubed is x squared times x squared times x squared, which is x times x, multiplied by x times x, multiplied by x times x. x times x has two factors. Two plus two plus two is six factors.
So we have x to the sixth power.
Notice that 6 is the product of the exponents, 2 and 3. x squared cubed is x to the power of 2 times 3, or x to the sixth power.

We write:

(x2)3x2·3x6(x2)3x2·3x6

We multiplied the exponents. This leads to the Power Property for Exponents.

Power Property for Exponents

If aa is a real number, and mandnmandn are whole numbers, then

(am)n=am·n(am)n=am·n

To raise a power to a power, multiply the exponents.

An example with numbers helps to verify this property.

(32)3=?32·3(9)3=?36729=729(32)3=?32·3(9)3=?36729=729

Example 6.22

Simplify: (y5)9(y5)9 (44)7.(44)7.

Try It 6.43

Simplify: (b7)5(b7)5 (54)3.(54)3.

Try It 6.44

Simplify: (z6)9(z6)9 (37)7.(37)7.

Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

(2x)3(2x)3
What does this mean? 2x·2x·2x2x·2x·2x
We group the like factors together. 2·2·2·x·x·x2·2·2·x·x·x
How many factors of 2 and of xx? 23·x323·x3

Notice that each factor was raised to the power and (2x)3(2x)3 is 23·x323·x3.

We write: (2x)3(2x)3
23·x323·x3

The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

Product to a Power Property for Exponents

If aa and bb are real numbers and mm is a whole number, then

(ab)m=ambm(ab)m=ambm

To raise a product to a power, raise each factor to that power.

An example with numbers helps to verify this property:

(2·3)2=?22·3262=?4·936=36(2·3)2=?22·3262=?4·936=36

Example 6.23

Simplify: (−9d)2(−9d)2 (3mn)3.(3mn)3.

Try It 6.45

Simplify: (−12y)2(−12y)2 (2wx)5.(2wx)5.

Try It 6.46

Simplify: (5wx)3(5wx)3 (−3y)3.(−3y)3.

Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

Properties of Exponents

If aandbaandb are real numbers, and mandnmandn are whole numbers, then

Product Property am·an=am+nam·an=am+n
Power Property (am)n=am·n(am)n=am·n
Product to a Power (ab)m=ambm(ab)m=ambm

All exponent properties hold true for any real numbers mandnmandn. Right now, we only use whole number exponents.

Example 6.24

Simplify: (y3)6(y5)4(y3)6(y5)4 (−6x4y5)2.(−6x4y5)2.

Try It 6.47

Simplify: (a4)5(a7)4(a4)5(a7)4 (−2c4d2)3.(−2c4d2)3.

Try It 6.48

Simplify: (−3x6y7)4(−3x6y7)4 (q4)5(q3)3.(q4)5(q3)3.

Example 6.25

Simplify: (5m)2(3m3)(5m)2(3m3) (3x2y)4(2xy2)3.(3x2y)4(2xy2)3.

Try It 6.49

Simplify: (5n)2(3n10)(5n)2(3n10) (c4d2)5(3cd5)4.(c4d2)5(3cd5)4.

Try It 6.50

Simplify: (a3b2)6(4ab3)4(a3b2)6(4ab3)4 (2x)3(5x7).(2x)3(5x7).

Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties of exponents to multiply monomials.

Example 6.26

Multiply: (3x2)(−4x3).(3x2)(−4x3).

Try It 6.51

Multiply: (5y7)(−7y4).(5y7)(−7y4).

Try It 6.52

Multiply: (−6b4)(−9b5).(−6b4)(−9b5).

Example 6.27

Multiply: (56x3y)(12xy2).(56x3y)(12xy2).

Try It 6.53

Multiply: (25a4b3)(15ab3).(25a4b3)(15ab3).

Try It 6.54

Multiply: (23r5s)(12r6s7).(23r5s)(12r6s7).

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with using multiplication properties of exponents:

Section 6.2 Exercises

Practice Makes Perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

88.

3535 9191 (13)2(13)2 (0.2)4(0.2)4

89.

104104 171171 (29)2(29)2 (0.5)3(0.5)3

90.

2626 141141 (25)3(25)3 (0.7)2(0.7)2

91.

8383 8181 (34)3(34)3 (0.4)3(0.4)3

92.

(−6)4(−6)4 6464

93.

(−2)6(−2)6 2626

94.

(14)4(14)4 (14)4(14)4

95.

(23)2(23)2 (23)2(23)2

96.

0.520.52 (−0.5)2(−0.5)2

97.

0.140.14 (−0.1)4(−0.1)4

Simplify Expressions Using the Product Property for Exponents

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

98.

d3·d6d3·d6

99.

x4·x2x4·x2

100.

n19·n12n19·n12

101.

q27·q15q27·q15

102.

45·4945·49 89·889·8

103.

310·36310·36 5·545·54

104.

y·y3y·y3 z25·z8z25·z8

105.

w5·ww5·w u41·u53u41·u53

106.

w·w2·w3w·w2·w3

107.

y·y3·y5y·y3·y5

108.

a4·a3·a9a4·a3·a9

109.

c5·c11·c2c5·c11·c2

110.

mx·m3mx·m3

111.

ny·n2ny·n2

112.

ya·ybya·yb

113.

xp·xqxp·xq

Simplify Expressions Using the Power Property for Exponents

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

114.

(m4)2(m4)2 (103)6(103)6

115.

(b2)7(b2)7 (38)2(38)2

116.

(y3)x(y3)x (5x)y(5x)y

117.

(x2)y(x2)y (7a)b(7a)b

Simplify Expressions Using the Product to a Power Property

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

118.

(6a)2(6a)2 (3xy)2(3xy)2

119.

(5x)2(5x)2 (4ab)2(4ab)2

120.

(−4m)3(−4m)3 (5ab)3(5ab)3

121.

(−7n)3(−7n)3 (3xyz)4(3xyz)4

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

122.

(y2)4·(y3)2(y2)4·(y3)2 (10a2b)3(10a2b)3

123.

(w4)3·(w5)2(w4)3·(w5)2 (2xy4)5(2xy4)5

124.

(−2r3s2)4(−2r3s2)4 (m5)3·(m9)4(m5)3·(m9)4

125.

(−10q2p4)3(−10q2p4)3 (n3)10·(n5)2(n3)10·(n5)2

126.

(3x)2(5x)(3x)2(5x) (5t2)3(3t)2(5t2)3(3t)2

127.

(2y)3(6y)(2y)3(6y) (10k4)3(5k6)2(10k4)3(5k6)2

128.

(5a)2(2a)3(5a)2(2a)3 (12y2)3(23y)2(12y2)3(23y)2

129.

(4b)2(3b)3(4b)2(3b)3 (12j2)5(25j3)2(12j2)5(25j3)2

130.

(25x2y)3(25x2y)3 (89xy4)2(89xy4)2

131.

(2r2)3(4r)2(2r2)3(4r)2 (3x3)3(x5)4(3x3)3(x5)4

132.

(m2n)2(2mn5)4(m2n)2(2mn5)4 (3pq4)2(6p6q)2(3pq4)2(6p6q)2

Multiply Monomials

In the following exercises, multiply the monomials.

133.

(6y7)(−3y4)(6y7)(−3y4)

134.

(−10x5)(−3x3)(−10x5)(−3x3)

135.

(−8u6)(−9u)(−8u6)(−9u)

136.

(−6c4)(−12c)(−6c4)(−12c)

137.

(15f8)(20f3)(15f8)(20f3)

138.

(14d5)(36d2)(14d5)(36d2)

139.

(4a3b)(9a2b6)(4a3b)(9a2b6)

140.

(6m4n3)(7mn5)(6m4n3)(7mn5)

141.

(47rs2)(14rs3)(47rs2)(14rs3)

142.

(58x3y)(24x5y)(58x3y)(24x5y)

143.

(23x2y)(34xy2)(23x2y)(34xy2)

144.

(35m3n2)(59m2n3)(35m3n2)(59m2n3)

Mixed Practice

In the following exercises, simplify each expression.

145.

(x2)4·(x3)2(x2)4·(x3)2

146.

(y4)3·(y5)2(y4)3·(y5)2

147.

(a2)6·(a3)8(a2)6·(a3)8

148.

(b7)5·(b2)6(b7)5·(b2)6

149.

(2m6)3(2m6)3

150.

(3y2)4(3y2)4

151.

(10x2y)3(10x2y)3

152.

(2mn4)5(2mn4)5

153.

(−2a3b2)4(−2a3b2)4

154.

(−10u2v4)3(−10u2v4)3

155.

(23x2y)3(23x2y)3

156.

(79pq4)2(79pq4)2

157.

(8a3)2(2a)4(8a3)2(2a)4

158.

(5r2)3(3r)2(5r2)3(3r)2

159.

(10p4)3(5p6)2(10p4)3(5p6)2

160.

(4x3)3(2x5)4(4x3)3(2x5)4

161.

(12x2y3)4(4x5y3)2(12x2y3)4(4x5y3)2

162.

(13m3n2)4(9m8n3)2(13m3n2)4(9m8n3)2

163.

(3m2n)2(2mn5)4(3m2n)2(2mn5)4

164.

(2pq4)3(5p6q)2(2pq4)3(5p6q)2

Everyday Math

165.

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 102102, on the third round is 103103, 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 102102
3 103103
6 ?
166.

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 $35,000(1.03)$35,000(1.03), after 2 years was $35,000(1.03)2$35,000(1.03)2, after 3 years was $35,000(1.03)3$35,000(1.03)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 $35,000(1.03)$35,000(1.03)
2 $35,000(1.03)2$35,000(1.03)2
3 $35,000(1.03)3$35,000(1.03)3
10 ?
167.

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 $1,000(0.70)$1,000(0.70) and the cost of the item during the second week would be $1,000(0.70)2$1,000(0.70)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 $1,000(0.70)$1,000(0.70)
2 $1,000(0.70)2$1,000(0.70)2
3
4
5 ?
168.

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 $20,000(0.90)$20,000(0.90) and the value of the car after the end of the second year would be $20,000(0.90)2$20,000(0.90)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.

Year Cost
1 $20,000(0.90)$20,000(0.90)
2 $20,000(0.90)2$20,000(0.90)2
3
8 ?

Writing Exercises

169.

Use the Product Property for Exponents to explain why x·x=x2x·x=x2.

170.

Explain why 53=(−5)353=(−5)3 but 54(−5)454(−5)4.

171.

Jorge thinks (12)2(12)2 is 1. What is wrong with his reasoning?

172.

Explain why x3·x5x3·x5 is x8x8, and not x15x15.

Self Check

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

This is a table that has seven rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “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,” and “multiply monomials.” The rest of the cells are blank.

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

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