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Intermediate Algebra 2e

8.3 Simplify Rational Exponents

Intermediate Algebra 2e8.3 Simplify Rational Exponents
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 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
    12. Chapter 12
  15. Index

Learning Objectives

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

  • Simplify expressions with a1na1n
  • Simplify expressions with amnamn
  • Use the properties of exponents to simplify expressions with rational exponents
Be Prepared 8.7

Before you get started, take this readiness quiz.

Add: 715+512.715+512.
If you missed this problem, review Example 1.28.

Be Prepared 8.8

Simplify: (4x2y5)3.(4x2y5)3.
If you missed this problem, review Example 5.18.

Be Prepared 8.9

Simplify: 5−3.5−3.
If you missed this problem, review Example 5.14.

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=81Since the bases are the same, the exponents must be equal.3p=1Solve forp.p=13(8p)3=8Multiply the exponents on the left.83p=8Write the exponent 1 on the right.83p=81Since the bases are the same, the exponents must be equal.3p=1Solve forp.p=13

So (813)3=8.(813)3=8. But we know also (83)3=8.(83)3=8. Then it must be that 813=83.813=83.

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

Rational Exponent a1na1n

If anan is a real number and n2,n2, then

a1n=ana1n=an

The denominator of the rational exponent is the index of the radical.

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 8.26

Write as a radical expression: x12x12 y13y13 z14.z14.

Try It 8.51

Write as a radical expression: t12t12 m13m13 r14.r14.

Try It 8.52

Write as a radial expression: b16b16 z15z15 p14.p14.

In the next example, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.

Example 8.27

Write with a rational exponent: 5y5y 4x34x3 35z4.35z4.

Try It 8.53

Write with a rational exponent: 10m10m 3n53n5 36y4.36y4.

Try It 8.54

Write with a rational exponent: 3k73k7 5j45j4 82a3.82a3.

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

Example 8.28

Simplify: 25122512 64136413 25614.25614.

Try It 8.55

Simplify: 36123612 813813 1614.1614.

Try It 8.56

Simplify: 1001210012 27132713 8114.8114.

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 8.29

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

Try It 8.57

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

Try It 8.58

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

Simplify Expressions with amnamn

We can look at amnamn in two ways. Remember the Power Property tells us to multiply the exponents and so (a1n)m(a1n)m and (am)1n(am)1n both equal amn.amn. If we write these expressions in radical form, we get

amn=(a1n)m=(an)mandamn=(am)1n=amnamn=(a1n)m=(an)mandamn=(am)1n=amn

This leads us to the following definition.

Rational Exponent amnamn

For any positive integers m and n,

amn=(an)mandamn=amnamn=(an)mandamn=amn

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, before raising it to the power indicated.

Example 8.30

Write with a rational exponent: y3y3 (2x3)4(2x3)4 (3a4b)3.(3a4b)3.

Try It 8.59

Write with a rational exponent: x5x5 (3y4)3(3y4)3 (2m3n)5.(2m3n)5.

Try It 8.60

Write with a rational exponent: a25a25 (5ab3)5(5ab3)5 (7xyz)3.(7xyz)3.

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

Example 8.31

Simplify: 1252312523 16321632 3225.3225.

Try It 8.61

Simplify: 27232723 81328132 1634.1634.

Try It 8.62

Simplify: 432432 27232723 62534.62534.

Example 8.32

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

Try It 8.63

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

Try It 8.64

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

Use the Properties of Exponents to Simplify Expressions with Rational Exponents

The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponenets here to have them for reference as we simplify expressions.

Properties of Exponents

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

Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmQuotient Propertyaman=amn,a0Zero Exponent Definitiona0=1,a0Quotient to a Power Property(ab)m=ambm,b0Negative Exponent Propertyan=1an,a0Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmQuotient Propertyaman=amn,a0Zero Exponent Definitiona0=1,a0Quotient to a Power Property(ab)m=ambm,b0Negative Exponent Propertyan=1an,a0

We will apply these properties in the next example.

Example 8.33

Simplify: x12·x56x12·x56 (z9)23(z9)23 x13x53.x13x53.

Try It 8.65

Simplify: x16·x43x16·x43 (x6)43(x6)43 x23x53.x23x53.

Try It 8.66

Simplify: y34·y58y34·y58 (m9)29(m9)29 d15d65.d15d65.

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

Example 8.34

Simplify: (27u12)23(27u12)23 (m23n12)32.(m23n12)32.

Try It 8.67

Simplify: (32x13)35(32x13)35 (x34y12)23.(x34y12)23.

Try It 8.68

Simplify: (81n25)32(81n25)32 (a32b12)43.(a32b12)43.

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

Example 8.35

Simplify: x34·x14x64x34·x14x64 (16x43y56x23y16)12.(16x43y56x23y16)12.

Try It 8.69

Simplify: m23·m13m53m23·m13m53 (25m16n116m23n16)12.(25m16n116m23n16)12.

Try It 8.70

Simplify: u45·u25u135u45·u25u135 (27x45y16x15y56)13.(27x45y16x15y56)13.

Media Access Additional Online Resources

Access these online resources for additional instruction and practice with simplifying rational exponents.

Section 8.3 Exercises

Practice Makes Perfect

Simplify expressions with a1na1n

In the following exercises, write as a radical expression.

119.

x12x12 y13y13 z14z14

120.

r12r12 s13s13 t14t14

121.

u15u15 v19v19 w120w120

122.

g17g17 h15h15 j125j125

In the following exercises, write with a rational exponent.

123.

x7x7 y9y9 f5f5

124.

r8r8 s10s10 t4t4

125.

7c37c3 12d712d7 26b426b4

126.

5x45x4 9y89y8 73z573z5

127.

21p21p 8q48q4 436r6436r6

128.

25a325a3 3b3b 40c840c8

In the following exercises, simplify.

129.

81128112 1251312513 64126412

130.

6251462514 2431524315 32153215

131.

16141614 16121612 6251462514

132.

64136413 32153215 81148114

133.

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

134.

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

135.

(−81)14(−81)14 81148114 (81)14(81)14

136.

(−49)12(−49)12 49124912 (49)12(49)12

137.

(−36)12(−36)12 36123612 (36)12(36)12

138.

(−16)14(−16)14 16141614 16141614

139.

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

140.

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

Simplify Expressions with amnamn

In the following exercises, write with a rational exponent.

141.

m5m5 (3y3)7(3y3)7 (4x5y)35(4x5y)35

142.

r74r74 (2pq5)3(2pq5)3 (12m7n)34(12m7n)34

143.

u25u25 (6x3)5(6x3)5 (18a5b)74(18a5b)74

144.

a3a3 (21v4)3(21v4)3 (2xy5z)24(2xy5z)24

In the following exercises, simplify.

145.

64526452 81−3281−32 (−27)23(−27)23

146.

25322532 932932 (−64)23(−64)23

147.

32253225 27232723 (−25)12(−25)12

148.

1003210032 49524952 (−100)32(−100)32

149.

932932 932932 (−9)32(−9)32

150.

64326432 64326432 (−64)32(−64)32

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

151.

c14·c58c14·c58 (p12)34(p12)34 r45r95r45r95

152.

652·612652·612 (b15)35(b15)35 w27w97w27w97

153.

y12·y34y12·y34 (x12)23(x12)23 m58m138m58m138

154.

q23·q56q23·q56 (h6)43(h6)43 n35n85n35n85

155.

(27q32)43(27q32)43 (a13b23)32(a13b23)32

156.

(64s37)16(64s37)16 (m43n12)34(m43n12)34

157.

(16u13)34(16u13)34 (4p13q12)32(4p13q12)32

158.

(625n83)34(625n83)34 (9x25y35)52(9x25y35)52

159.

r52·r12r32r52·r12r32 (36s15t32s95t12)12(36s15t32s95t12)12

160.

a34·a14a104a34·a14a104 (27b23c52b73c12)13(27b23c52b73c12)13

161.

c53·c13c23c53·c13c23 (8x53y1227x43y52)13(8x53y1227x43y52)13

162.

m74·m54m24m74·m54m24 (16m15n3281m95n12)14(16m15n3281m95n12)14

Writing Exercises

163.

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

164.

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

Self Check

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

This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions with a to the power of 1 divided by n.”, “simplify expression with a to the power of m divided by n”, and “use the laws of exponents to simplify expression with rational exponents”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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