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Intermediate Algebra

7.3 Simplify Complex Rational Expressions

Intermediate Algebra7.3 Simplify Complex Rational Expressions
  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 a complex rational expression by writing it as division
  • Simplify a complex rational expression by using the LCD
Be Prepared 7.3

Before you get started, take this readiness quiz.

  1. Simplify: 35910.35910.
    If you missed this problem, review Example 1.27.
  2. Simplify: 11342+4·5.11342+4·5.
    If you missed this problem, review Example 1.31.
  3. Solve: 12x+14=18.12x+14=18.
    If you missed this problem, review Example 2.9.

Simplify a Complex Rational Expression by Writing it as Division

Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously simplified complex fractions like these:

3458x2xy63458x2xy6

In this section, we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.

Complex Rational Expression

A complex rational expression is a rational expression in which the numerator and/or the denominator contains a rational expression.

Here are a few complex rational expressions:

4y38y291x+1yxyyx2x+64x64x2364y38y291x+1yxyyx2x+64x64x236

Remember, we always exclude values that would make any denominator zero.

We will use two methods to simplify complex rational expressions.

We have already seen this complex rational expression earlier in this chapter.

6x27x+24x82x28x+3x25x+66x27x+24x82x28x+3x25x+6

We noted that fraction bars tell us to divide, so rewrote it as the division problem:

(6x27x+24x8)÷(2x28x+3x25x+6).(6x27x+24x8)÷(2x28x+3x25x+6).

Then, we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.

This is one method to simplify complex rational expressions. We make sure the complex rational expression is of the form where one fraction is over one fraction. We then write it as if we were dividing two fractions.

Example 7.24

Simplify the complex rational expression by writing it as division: 6x43x216.6x43x216.

Try It 7.47

Simplify the complex rational expression by writing it as division: 2x213x+1.2x213x+1.

Try It 7.48

Simplify the complex rational expression by writing it as division: 1x27x+122x4.1x27x+122x4.

Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.

Example 7.25

Simplify the complex rational expression by writing it as division: 13+161213.13+161213.

Try It 7.49

Simplify the complex rational expression by writing it as division: 12+2356+112.12+2356+112.

Try It 7.50

Simplify the complex rational expression by writing it as division: 341318+56.341318+56.

We follow the same procedure when the complex rational expression contains variables.

Example 7.26

How to Simplify a Complex Rational Expression using Division

Simplify the complex rational expression by writing it as division: 1x+1yxyyx.1x+1yxyyx.

Try It 7.51

Simplify the complex rational expression by writing it as division: 1x+1y1x1y.1x+1y1x1y.

Try It 7.52

Simplify the complex rational expression by writing it as division: 1a+1b1a21b21a+1b1a21b2.

We summarize the steps here.

How To

Simplify a complex rational expression by writing it as division.

  1. Step 1. Simplify the numerator and denominator.
  2. Step 2. Rewrite the complex rational expression as a division problem.
  3. Step 3. Divide the expressions.

Example 7.27

Simplify the complex rational expression by writing it as division: n4nn+51n+5+1n5.n4nn+51n+5+1n5.

Try It 7.53

Simplify the complex rational expression by writing it as division: b3bb+52b+5+1b5.b3bb+52b+5+1b5.

Try It 7.54

Simplify the complex rational expression by writing it as division: 13c+41c+4+c3.13c+41c+4+c3.

Simplify a Complex Rational Expression by Using the LCD

We “cleared” the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by the LCD of all the rational expressions.

Let’s look at the complex rational expression we simplified one way in Example 7.25. We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by LCDLCDLCDLCD we are multiplying by 1, so the value stays the same.

Example 7.28

Simplify the complex rational expression by using the LCD: 13+161213.13+161213.

Try It 7.55

Simplify the complex rational expression by using the LCD: 12+15110+15.12+15110+15.

Try It 7.56

Simplify the complex rational expression by using the LCD: 14+3812516.14+3812516.

We will use the same example as in Example 7.26. Decide which method works better for you.

Example 7.29

How to Simplify a Complex Rational Expressing using the LCD

Simplify the complex rational expression by using the LCD: 1x+1yxyyx.1x+1yxyyx.

Try It 7.57

Simplify the complex rational expression by using the LCD: 1a+1bab+ba.1a+1bab+ba.

Try It 7.58

Simplify the complex rational expression by using the LCD: 1x21y21x+1y.1x21y21x+1y.

How To

Simplify a complex rational expression by using the LCD.

  1. Step 1. Find the LCD of all fractions in the complex rational expression.
  2. Step 2. Multiply the numerator and denominator by the LCD.
  3. Step 3. Simplify the expression.

Be sure to start by factoring all the denominators so you can find the LCD.

Example 7.30

Simplify the complex rational expression by using the LCD: 2x+64x64x236.2x+64x64x236.

Try It 7.59

Simplify the complex rational expression by using the LCD: 3x+25x23x24.3x+25x23x24.

Try It 7.60

Simplify the complex rational expression by using the LCD: 2x71x+76x+71x249.2x71x+76x+71x249.

Be sure to factor the denominators first. Proceed carefully as the math can get messy!

Example 7.31

Simplify the complex rational expression by using the LCD: 4m27m+123m32m4.4m27m+123m32m4.

Try It 7.61

Simplify the complex rational expression by using the LCD: 3x2+7x+104x+2+1x+5.3x2+7x+104x+2+1x+5.

Try It 7.62

Simplify the complex rational expression by using the LCD: 4yy+5+2y+63yy2+11y+30.4yy+5+2y+63yy2+11y+30.

Example 7.32

Simplify the complex rational expression by using the LCD: yy+11+1y1.yy+11+1y1.

Try It 7.63

Simplify the complex rational expression by using the LCD: xx+31+1x+3.xx+31+1x+3.

Try It 7.64

Simplify the complex rational expression by using the LCD: 1+1x13x+1.1+1x13x+1.

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with complex fractions.

Section 7.3 Exercises

Practice Makes Perfect

Simplify a Complex Rational Expression by Writing it as Division

In the following exercises, simplify each complex rational expression by writing it as division.

151.

2aa+44a2a2162aa+44a2a216

152.

3bb5b2b2253bb5b2b225

153.

5c2+5c1410c+75c2+5c1410c+7

154.

8d2+9d+1812d+68d2+9d+1812d+6

155.

12+5623+7912+5623+79

156.

12+3435+71012+3435+710

157.

231934+56231934+56

158.

121623+34121623+34

159.

nm+1n1nnmnm+1n1nnm

160.

1p+pqqp1q1p+pqqp1q

161.

1r+1t1r21t21r+1t1r21t2

162.

2v+2w1v21w22v+2w1v21w2

163.

x2xx+31x+3+1x3x2xx+31x+3+1x3

164.

y2yy42y4+2y+4y2yy42y4+2y+4

165.

22a+31a+3+a222a+31a+3+a2

166.

4+4b51b5+b44+4b51b5+b4

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify each complex rational expression by using the LCD.

167.

13+1814+11213+1814+112

168.

14+1916+11214+1916+112

169.

56+297181356+2971813

170.

16+415351216+4153512

171.

cd+1d1ddccd+1d1ddc

172.

1m+mnnm1n1m+mnnm1n

173.

1p+1q1p21q21p+1q1p21q2

174.

2r+2t1r21t22r+2t1r21t2

175.

2x+53x5+1x2252x+53x5+1x225

176.

5y43y+4+2y2165y43y+4+2y216

177.

5z264+3z+81z+8+2z85z264+3z+81z+8+2z8

178.

3s+6+5s61s236+4s+63s+6+5s61s236+4s+6

179.

4a22a151a5+2a+34a22a151a5+2a+3

180.

5b26b273b9+1b+35b26b273b9+1b+3

181.

5c+23c+75cc2+9c+145c+23c+75cc2+9c+14

182.

6d42d+72dd2+3d286d42d+72dd2+3d28

183.

2+1p35p32+1p35p3

184.

nn23+5n2nn23+5n2

185.

mm+54+1m5mm+54+1m5

186.

7+2q21q+27+2q21q+2

In the following exercises, simplify each complex rational expression using either method.

187.

342712+514342712+514

188.

vw+1v1vvwvw+1v1vvw

189.

2a+41a2162a+41a216

190.

3b23b405b+52b83b23b405b+52b8

191.

3m+3n1m21n23m+3n1m21n2

192.

2r91r+9+3r2812r91r+9+3r281

193.

x3xx+23x+2+3x2x3xx+23x+2+3x2

194.

yy+32+1y3yy+32+1y3

Writing Exercises

195.

In this section, you learned to simplify the complex fraction 3x+2xx243x+2xx24 two ways: rewriting it as a division problem or multiplying the numerator and denominator by the LCD. Which method do you prefer? Why?

196.

Efraim wants to start simplifying the complex fraction 1a+1b1a1b1a+1b1a1b by cancelling the variables from the numerator and denominator, 1a+1b1a1b.1a+1b1a1b. Explain what is wrong with Efraim’s plan.

Self Check

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

This table has four columns and three rows. The first row is a header and it labels each column, “I can…”, “Confidently,” “With some help,” and “No-I don’t get it!” In row 2, the I can was simplify a complex rational expression by writing it as division. In row 3, the I can was simplify a complex rational expression by using the least common denominator.

After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

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