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Elementary Algebra

9.8 Rational Exponents

Elementary Algebra9.8 Rational 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 Solve 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 Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    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
    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 a1na1n
  • Simplify expressions with amnamn
  • Use the Laws of Exponents to simply expressions with rational exponents
Be Prepared 9.8

Before you get started, take this readiness quiz.

  1. Add: 715+512715+512.
    If you missed this problem, review Example 1.81.
  2. Simplify: (4x2y5)3(4x2y5)3.
    If you missed this problem, review Example 6.24.
  3. Simplify: 5−35−3.
    If you missed this problem, review Example 6.89.

Simplify Expressions with a1na1n

Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.

The Power Property for Exponents says that (am)n=am·n(am)n=am·n when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.

Suppose we want to find a number p such that (8p)3=8(8p)3=8. We will use the Power Property of Exponents to find the value of p.

(8p)3=8Multiply the exponents on the left.83p=8Write the exponent 1 on the right.83p=81The exponents must be equal.3p=1Solve forp.p=13So(813)3=8.(8p)3=8Multiply the exponents on the left.83p=8Write the exponent 1 on the right.83p=81The exponents must be equal.3p=1Solve forp.p=13So(813)3=8.

But we know also (83)3=8(83)3=8. Then it must be that 813=83813=83.

This same logic can be used for any positive integer exponent n to show that a1n=ana1n=an.

Rational Exponent a1na1n

If anan is a real number and n2n2, a1n=ana1n=an.

There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.

Example 9.103

Write as a radical expression: x12x12 y13y13 z14z14.

Try It 9.205

Write as a radical expression: t12t12 m13m13 r14r14.

Try It 9.206

Write as a radial expression: b12b12 z13z13 p14p14.

Example 9.104

Write with a rational exponent: xx y3y3 z4z4.

Try It 9.207

Write with a rational exponent: ss x3x3 b4b4.

Try It 9.208

Write with a rational exponent: vv p3p3 p4p4.

Example 9.105

Write with a rational exponent: 5y5y 4x34x3 35z435z4.

Try It 9.209

Write with a rational exponent: 10m10m 3n53n5 36y436y4.

Try It 9.210

Write with a rational exponent: 3k73k7 5j45j4 82a382a3.

In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.

Example 9.106

Simplify: 25122512 64136413 2561425614.

Try It 9.211

Simplify: 36123612 813813 16141614.

Try It 9.212

Simplify: 1001210012 27132713 81148114.

Be careful of the placement of the negative signs in the next example. We will need to use the property an=1anan=1an in one case.

Example 9.107

Simplify: (−64)13(−64)13 64136413 (64)13(64)13.

Try It 9.213

Simplify: (−125)13(−125)13 1251312513 (125)13(125)13.

Try It 9.214

Simplify: (−32)15(−32)15 32153215 (32)15(32)15.

Example 9.108

Simplify: (−16)14(−16)14 16141614 (16)14(16)14.

Try It 9.215

Simplify: (−64)12(−64)12 64126412 (64)12(64)12.

Try It 9.216

Simplify: (−256)14(−256)14 2561425614 (256)14(256)14.

Simplify Expressions with amnamn

Let’s work with the Power Property for Exponents some more.

Suppose we raise a1na1n to the power m.

(a1n)mMultiply the exponents.a1n·mSimplify.amnSoamn=(an)m.(a1n)mMultiply the exponents.a1n·mSimplify.amnSoamn=(an)m.

Now suppose we take amam to the 1n1n power.

(am)1nMultiply the exponents.am·1nSimplify.amnSoamn=amnalso.(am)1nMultiply the exponents.am·1nSimplify.amnSoamn=amnalso.

Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller.

Rational Exponent amnamn

For any positive integers m and n,

amn=(an)mamn=amnamn=(an)mamn=amn

Example 9.109

Write with a rational exponent: y3y3 x23x23 z34z34.

Try It 9.217

Write with a rational exponent: x5x5 z34z34 y25y25.

Try It 9.218

Write with a rational exponent: a25a25 b73b73 m54m54.

Example 9.110

Simplify: 932932 1252312523 81348134.

Try It 9.219

Simplify: 432432 27232723 6253462534.

Try It 9.220

Simplify: 853853 81328132 16341634.

Remember that bp=1bpbp=1bp. The negative sign in the exponent does not change the sign of the expression.

Example 9.111

Simplify: 16321632 32253225 452452.

Try It 9.221

Simplify: 853853 81328132 16341634.

Try It 9.222

Simplify: 432432 27232723 6253462534.

Example 9.112

Simplify: 25322532 25322532 (−25)32(−25)32.

Try It 9.223

Simplify: −1632−1632 −1632−1632 (−16)32(−16)32.

Try It 9.224

Simplify: −8132−8132 −8132−8132 (−81)32(−81)32.

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

The same laws of exponents that we already used apply to rational exponents, too. We will list the Exponent Properties here to have them for reference as we simplify expressions.

Summary of Exponent Properties

If a,ba,b are real numbers and m,nm,n are rational numbers, then

Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmQuotient Propertyaman=amn,a0,m>naman=1anm,a0,n>mZero Exponent Definitiona0=1,a0Quotient to a Power Property(ab)m=ambm,b0Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmQuotient Propertyaman=amn,a0,m>naman=1anm,a0,n>mZero Exponent Definitiona0=1,a0Quotient to a Power Property(ab)m=ambm,b0

When we multiply the same base, we add the exponents.

Example 9.113

Simplify: 212·252212·252 x23·x43x23·x43 z34·z54z34·z54.

Try It 9.225

Simplify: 323·343323·343 y13·y83y13·y83 m14·m34m14·m34.

Try It 9.226

Simplify: 535·575535·575 z18·z78z18·z78 n27·n57n27·n57.

We will use the Power Property in the next example.

Example 9.114

Simplify: (x4)12(x4)12 (y6)13(y6)13 (z9)23(z9)23.

Try It 9.227

Simplify: (p10)15(p10)15 (q8)34(q8)34 (x6)43(x6)43.

Try It 9.228

Simplify: (r6)53(r6)53 (s12)34(s12)34 (m9)29(m9)29.

The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

Example 9.115

Simplify: x43x13x43x13 y34y14y34y14 z23z53z23z53.

Try It 9.229

Simplify: u54u14u54u14 v35v25v35v25 x23x53x23x53.

Try It 9.230

Simplify: c125c25c125c25 m54m94m54m94 d15d65d15d65.

Sometimes we need to use more than one property. In the next two examples, we will use both the Product to a Power Property and then the Power Property.

Example 9.116

Simplify: (27u12)23(27u12)23 (8v14)23(8v14)23.

Try It 9.231

Simplify: (32x13)35(32x13)35 (64y23)13(64y23)13.

Try It 9.232

Simplify: (16m13)32(16m13)32 (81n25)32(81n25)32.

Example 9.117

Simplify: (m3n9)13(m3n9)13 (p4q8)14(p4q8)14.

We will use both the Product and Quotient Properties in the next example.

Example 9.118

Simplify: x34·x14x64x34·x14x64 y43·yy23y43·yy23.

Try It 9.233

Simplify: m23·m13m53m23·m13m53 n16·nn116n16·nn116.

Try It 9.234

Simplify: u45·u25u135u45·u25u135 v12·vv72v12·vv72.

Section 9.8 Exercises

Practice Makes Perfect

Simplify Expressions with a1na1n

In the following exercises, write as a radical expression.

524.


x12x12
y13y13
z14z14

525.


r12r12
s13s13
t14t14

526.


u15u15
v19v19
w120w120

527.


g17g17
h15h15
j125j125

In the following exercises, write with a rational exponent.

528.


x7x7
y9y9
f5f5

529.


r8r8

.


t4t4

530.


a3a3

.


cc

531.


u5u5
vv

.
532.


7c37c3
12d712d7
35f435f4

533.


5x45x4
9y89y8
73z573z5

534.


21p21p
8q48q4
436r6436r6

535.


25a325a3
3b3b

.

In the following exercises, simplify.

536.


81128112
1251312513
64126412

537.


6251462514
2431524315
32153215

538.


16141614
16121612
312515312515

539.


2161321613
32153215
81148114

540.


(−216)13(−216)13
2161321613
(216)13(216)13

541.


(−243)15(−243)15
2431524315
(243)15(243)15

542.


(−1)13(−1)13
−113−113
(1)13(1)13

543.


(−1000)13(−1000)13
100013100013
(1000)13(1000)13

544.


(−81)14(−81)14
81148114
(81)14(81)14

545.


(−49)12(−49)12
49124912
(49)12(49)12

546.


(−36)12(−36)12
36123612
(36)12(36)12

547.


(−1)14(−1)14
(1)14(1)14
114114

548.


(−100)12(−100)12
1001210012
(100)12(100)12

549.


(−32)15(−32)15
(243)15(243)15
1251312513

Simplify Expressions with amnamn

In the following exercises, write with a rational exponent.

550.


m5m5
n23n23
p34p34

551.


r74r74
s35s35
t73t73

552.


u25u25
v85v85
w49w49

553.


a3a3
b5b5
c53c53

In the following exercises, simplify.

554.


16321632
823823
10,0003410,00034

555.


100023100023
25322532
32353235

556.


27532753
16541654
32253225

557.


16321632
1255312553
64436443

558.


32253225
27232723
25322532

559.


64526452
81328132
27432743

560.


25322532
932932
(−64)23(−64)23

561.


1003210032
49524952
(−100)32(−100)32

562.


932932
932932
(−9)32(−9)32

563.


64326432
64326432
(−64)32(−64)32

564.


1003210032
1003210032
(−100)32(−100)32

565.


49324932
49324932
(−49)32(−49)32

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

566.


458·4118458·4118
m712·m1712m712·m1712
p37·p187p37·p187

567.


652·612652·612
n210·n810n210·n810
q25·q135q25·q135

568.


512·572512·572
c34·c94c34·c94
d35·d25d35·d25

569.


1013·10531013·1053
x56·x76x56·x76
y118·y218y118·y218

570.


(m6)52(m6)52
(n9)43(n9)43
(p12)34(p12)34

571.


(a12)16(a12)16
(b15)35(b15)35
(c11)111(c11)111

572.


(x12)23(x12)23
(y20)25(y20)25
(z16)116(z16)116

573.


(h6)43(h6)43
(k12)34(k12)34
(j10)75(j10)75

574.


x72x52x72x52
y52y12y52y12
r45r95r45r95

575.


s115s65s115s65
z73z13z73z13
w27w97w27w97

576.


t125t75t125t75
x32x12x32x12
m138m58m138m58

577.


u139u49u139u49
r157r87r157r87
n35n85n35n85

578.


(9p23)52(9p23)52
(27q32)43(27q32)43

579.


(81r45)14(81r45)14
(64s37)16(64s37)16

580.


(16u13)34(16u13)34
(100v25)32(100v25)32

581.


(27m34)23(27m34)23
(625n83)34(625n83)34

582.


(x8y10)12(x8y10)12
(a9b12)13(a9b12)13

583.


(r8s4)14(r8s4)14
(u15v20)15(u15v20)15

584.


(a6b16)12(a6b16)12
(j9k6)23(j9k6)23

585.


(r16s10)12(r16s10)12
(u10v5)45(u10v5)45

586.


r52·r12r32r52·r12r32
s15·ss95s15·ss95

587.


a34·a14a104a34·a14a104
b23·bb73b23·bb73

588.


c53·c13c23c53·c13c23
d35·dd25d35·dd25

589.


m74·m54m24m74·m54m24
n37·nn47n37·nn47

590.

452·412452·412

591.

n26·n46n26·n46

592.

(a24)16(a24)16

593.

(b10)35(b10)35

594.

w25w75w25w75

595.

z23z83z23z83

596.

(27r35)13(27r35)13

597.

(64s35)16(64s35)16

598.

(r9s12)13(r9s12)13

599.

(u12v18)16(u12v18)16

Everyday Math

600.

Landscaping Joe wants to have a square garden plot in his backyard. He has enough compost to cover an area of 144 square feet. Simplify 1441214412 to find the length of each side of his garden.

601.

Landscaping Elliott wants to make a square patio in his yard. He has enough concrete to pave an area of 242 square feet. Simplify 2421224212 to find the length of each side of his patio.Round to the nearest tenth of a foot.

602.

Gravity While putting up holiday decorations, Bob dropped a decoration from the top of a tree that is 12 feet tall. Simplify 1212161212121612 to find how many seconds it took for the decoration to reach the ground. Round to the nearest tenth of a second.

603.

Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Simplify 10241216121024121612 to find how many seconds it took for the flare to reach the water.

Writing Exercises

604.

Show two different algebraic methods to simplify 432.432. Explain all your steps.

605.

Explain why the expression (−16)32(−16)32 cannot be evaluated.

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