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Prealgebra 2e

10.2 Use Multiplication Properties of Exponents

Prealgebra 2e10.2 Use Multiplication Properties of Exponents
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. 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
    11. Chapter 11
  17. Index

Learning Objectives

By the end of this section, you will be able to:
  • Simplify expressions with exponents
  • Simplify expressions using the Product Property of Exponents
  • Simplify expressions using the Power Property of Exponents
  • Simplify expressions using the Product to a Power Property
  • Simplify expressions by applying several properties
  • Multiply monomials
Be Prepared 10.4

Before you get started, take this readiness quiz.

Simplify: 34·34.34·34.
If you missed the problem, review Example 4.25.

Be Prepared 10.5

Simplify: (−2)(−2)(−2).(−2)(−2)(−2).
If you missed the problem, review Example 3.52.

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, 2424 means to multiply four factors of 2,2, so 2424 means 2·2·2·2.2·2·2·2. This format is known as exponential notation.

Exponential Notation

On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.

This is read aa to the mthmth power.

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

On the left side, 7 to the 3rd power is shown. Below is 7 times 7 times 7, with 3 factors written below. On the right side, parentheses negative 8 to the 5th power is shown. Below is negative 8 times negative 8 times negative 8 times negative 8 times negative 8, with 5 factors written below.

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

Example 10.11

Simplify:

  1. 5353
  2. 9191
Try It 10.21

Simplify:

  1. 4343
  2. 111111
Try It 10.22

Simplify:

  1. 3434
  2. 211211

Example 10.12

Simplify:

  1. (78)2(78)2
  2. (0.74)2(0.74)2
Try It 10.23

Simplify:

  1. (58)2(58)2
  2. (0.67)2(0.67)2
Try It 10.24

Simplify:

  1. (25)3(25)3
  2. (0.127)2(0.127)2

Example 10.13

Simplify:

  1. (−3)4(−3)4
  2. −34−34
Try It 10.25

Simplify:

  1. (−2)4(−2)4
  2. −24−24
Try It 10.26

Simplify:

  1. (−8)2(−8)2
  2. −82−82

Simplify Expressions Using the Product Property of 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. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.

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

.
What does this mean?

How many factors altogether?
.
So, we have .
Notice that 5 is the sum of the exponents, 2 and 3. .
We write: x2x3x2x3
x2+3x2+3
x5x5

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

Product Property of Exponents

If aa is a real number and m,nm,n 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 10.14

Simplify: x5·x7.x5·x7.

Try It 10.27

Simplify: x7·x8.x7·x8.

Try It 10.28

Simplify: x5·x11.x5·x11.

Example 10.15

Simplify: b4·b.b4·b.

Try It 10.29

Simplify: p9·p.p9·p.

Try It 10.30

Simplify: m·m7.m·m7.

Example 10.16

Simplify: 27·29.27·29.

Try It 10.31

Simplify: 6·69.6·69.

Try It 10.32

Simplify: 96·99.96·99.

Example 10.17

Simplify: y17·y23.y17·y23.

Try It 10.33

Simplify: y24·y19.y24·y19.

Try It 10.34

Simplify: z15·z24.z15·z24.

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

Example 10.18

Simplify: x3·x4·x2.x3·x4·x2.

Try It 10.35

Simplify: x7·x5·x9.x7·x5·x9.

Try It 10.36

Simplify: y3·y8·y4.y3·y8·y4.

Simplify Expressions Using the Power Property of 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.

.
.
What does this mean?

How many factors altogether?
.
So, we have .
Notice that 6 is the product of the exponents, 2 and 3. .
We write: (x2)3(x2)3
x23x23
x6x6

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

Power Property of Exponents

If aa is a real number and m,nm,n 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.

(52)3=?52·3(25)3=?5615,625=15,625(52)3=?52·3(25)3=?5615,625=15,625

Example 10.19

Simplify:

  1. (x5)7(x5)7
  2. (36)8(36)8
Try It 10.37

Simplify:

  1. (x7)4(x7)4
  2. (74)8(74)8
Try It 10.38

Simplify:

  1. (x6)9(x6)9
  2. (86)7(86)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. Look for a 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 x?x? 23·x323·x3
Notice that each factor was raised to the power. (2x)3is23·x3(2x)3is23·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 of 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 10.20

Simplify: (−11x)2.(−11x)2.

Try It 10.39

Simplify: (−14x)2.(−14x)2.

Try It 10.40

Simplify: (−12a)2.(−12a)2.

Example 10.21

Simplify: (3xy)3.(3xy)3.

Try It 10.41

Simplify: (−4xy)4.(−4xy)4.

Try It 10.42

Simplify: (6xy)3.(6xy)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 a,ba,b are real numbers and m,nm,n are whole numbers, then

Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power Property(ab)m=ambmProduct Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power Property(ab)m=ambm

Example 10.22

Simplify: (x2)6(x5)4.(x2)6(x5)4.

Try It 10.43

Simplify: (x4)3(x7)4.(x4)3(x7)4.

Try It 10.44

Simplify: (y9)2(y8)3.(y9)2(y8)3.

Example 10.23

Simplify: (7x3y4)2.(7x3y4)2.

Try It 10.45

Simplify: (8x4y7)3.(8x4y7)3.

Try It 10.46

Simplify: (3a5b6)4.(3a5b6)4.

Example 10.24

Simplify: (6n)2(4n3).(6n)2(4n3).

Try It 10.47

Simplify: (7n)2(2n12).(7n)2(2n12).

Try It 10.48

Simplify: (4m)2(3m3).(4m)2(3m3).

Example 10.25

Simplify: (3p2q)4(2pq2)3.(3p2q)4(2pq2)3.

Try It 10.49

Simplify: (u3v2)5(4uv4)3.(u3v2)5(4uv4)3.

Try It 10.50

Simplify: (5x2y3)2(3xy4)3.(5x2y3)2(3xy4)3.

Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

Example 10.26

Multiply: (4x2)(5x3).(4x2)(5x3).

Try It 10.51

Multiply: (7x7)(8x4).(7x7)(8x4).

Try It 10.52

Multiply: (9y4)(6y5).(9y4)(6y5).

Example 10.27

Multiply: (34c3d)(12cd2).(34c3d)(12cd2).

Try It 10.53

Multiply: (45m4n3)(15mn3).(45m4n3)(15mn3).

Try It 10.54

Multiply: (23p5q)(18p6q7).(23p5q)(18p6q7).

Media Access Additional Online Resources

Section 10.2 Exercises

Practice Makes Perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

55.

4545

56.

103103

57.

(12)2(12)2

58.

(35)2(35)2

59.

(0.2)3(0.2)3

60.

(0.4)3(0.4)3

61.

(−5)4(−5)4

62.

(−3)5(−3)5

63.

−54−54

64.

−35−35

65.

−104−104

66.

−26−26

67.

(23)3(23)3

68.

(14)4(14)4

69.

0.520.52

70.

0.140.14

Simplify Expressions Using the Product Property of Exponents

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

71.

x3·x6x3·x6

72.

m4·m2m4·m2

73.

a·a4a·a4

74.

y12·yy12·y

75.

35·3935·39

76.

510·56510·56

77.

z·z2·z3z·z2·z3

78.

a·a3·a5a·a3·a5

79.

xa·x2xa·x2

80.

yp·y3yp·y3

81.

ya·ybya·yb

82.

xp·xqxp·xq

Simplify Expressions Using the Power Property of Exponents

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

83.

(u4)2(u4)2

84.

(x2)7(x2)7

85.

(y5)4(y5)4

86.

(a3)2(a3)2

87.

(102)6(102)6

88.

(28)3(28)3

89.

(x15)6(x15)6

90.

(y12)8(y12)8

91.

(x2)y(x2)y

92.

(y3)x(y3)x

93.

(5x)y(5x)y

94.

(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.

95.

(5a)2(5a)2

96.

(7x)2(7x)2

97.

(6m)3(6m)3

98.

(9n)3(9n)3

99.

(4rs)2(4rs)2

100.

(5ab)3(5ab)3

101.

(4xyz)4(4xyz)4

102.

(5abc)3(5abc)3

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

103.

(x2)4·(x3)2(x2)4·(x3)2

104.

(y4)3·(y5)2(y4)3·(y5)2

105.

(a2)6·(a3)8(a2)6·(a3)8

106.

(b7)5·(b2)6(b7)5·(b2)6

107.

(3x)2(5x)(3x)2(5x)

108.

(2y)3(6y)(2y)3(6y)

109.

(5a)2(2a)3(5a)2(2a)3

110.

(4b)2(3b)3(4b)2(3b)3

111.

(2m6)3(2m6)3

112.

(3y2)4(3y2)4

113.

(10x2y)3(10x2y)3

114.

(2mn4)5(2mn4)5

115.

(−2a3b2)4(−2a3b2)4

116.

(−10u2v4)3(−10u2v4)3

117.

(23x2y)3(23x2y)3

118.

(79pq4)2(79pq4)2

119.

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

120.

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

121.

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

122.

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

123.

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

124.

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

125.

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

126.

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

Multiply Monomials

In the following exercises, multiply the following monomials.

127.

(12x2)(−5x4)(12x2)(−5x4)

128.

(−10y3)(7y2)(−10y3)(7y2)

129.

(−8u6)(−9u)(−8u6)(−9u)

130.

(−6c4)(−12c)(−6c4)(−12c)

131.

(15r8)(20r3)(15r8)(20r3)

132.

(14a5)(36a2)(14a5)(36a2)

133.

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

134.

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

135.

(47xy2)(14xy3)(47xy2)(14xy3)

136.

(58u3v)(24u5v)(58u3v)(24u5v)

137.

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

138.

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

Everyday Math

139.

Email Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is 62,62, on the third round is 63,63, as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.

Round Number of people
11 66
22 6262
33 6363
88 ??
140.

Salary Raul’s boss gives him a 5%5% raise every year on his birthday. This means that each year, Raul’s salary is 1.051.05 times his last year’s salary. If his original salary was $40,000$40,000, his salary after 11 year was $40,000(1.05),$40,000(1.05), after 22 years was $40,000(1.05)2,$40,000(1.05)2, after 33 years was $40,000(1.05)3,$40,000(1.05)3, as shown in the table below. What will Raul’s salary be after 1010 years? Simplify the expression, to show Raul’s salary in dollars.

Year Salary
11 $40,000(1.05)$40,000(1.05)
22 $40,000(1.05)2$40,000(1.05)2
33 $40,000(1.05)3$40,000(1.05)3
1010 ??

Writing Exercises

141.

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

142.

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

143.

Jorge thinks (12)2(12)2 is 1.1. What is wrong with his reasoning?

144.

Explain why x3·x5x3·x5 is x8,x8, and not x15.x15.

Self Check

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

.

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

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