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Elementary Algebra

8.5 Simplify Complex Rational Expressions

Elementary Algebra8.5 Simplify Complex Rational Expressions
  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 a complex rational expression by writing it as division
  • Simplify a complex rational expression by using the LCD
Be Prepared 8.5

Before you get started, take this readiness quiz.

If you miss a problem, go back to the section listed and review the material.

  1. Simplify: 35910.35910.
    If you missed this problem, review Example 1.72.
  2. Simplify: 11342+4·5.11342+4·5.
    If you missed this problem, review Example 1.74.

Complex fractions are fractions in which the numerator or denominator contains a fraction. In Chapter 1 we 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 or 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.

Simplify a Complex Rational Expression by Writing it as Division

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 rational expressions. We write it as if we were dividing two fractions.

Example 8.50

Simplify: 4y38y29.4y38y29.

Try It 8.99

Simplify: 2x213x+1.2x213x+1.

Try It 8.100

Simplify: 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 8.51

Simplify: 13+161213.13+161213.

Try It 8.101

Simplify: 12+2356+112.12+2356+112.

Try It 8.102

Simplify: 341318+56.341318+56.

Example 8.52

How to Simplify a Complex Rational Expression by Writing it as Division

Simplify: 1x+1yxyyx.1x+1yxyyx.

Try It 8.103

Simplify: 1x+1y1x1y.1x+1y1x1y.

Try It 8.104

Simplify: 1a+1b1a21b2.1a+1b1a21b2.

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 8.53

Simplify: n4nn+51n+5+1n5.n4nn+51n+5+1n5.

Try It 8.105

Simplify: b3bb+52b+5+1b5.b3bb+52b+5+1b5.

Try It 8.106

Simplify: 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 LCD of all the rational expressions.

Let’s look at the complex rational expression we simplified one way in Example 8.51. 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 8.54

Simplify: 13+161213.13+161213.

Try It 8.107

Simplify: 12+15110+15.12+15110+15.

Try It 8.108

Simplify: 14+3812516.14+3812516.

Example 8.55

How to Simplify a Complex Rational Expression by Using the LCD

Simplify: 1x+1yxyyx.1x+1yxyyx.

Try It 8.109

Simplify: 1a+1bab+ba.1a+1bab+ba.

Try It 8.110

Simplify: 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 8.56

Simplify: 2x+64x64x236.2x+64x64x236.

Try It 8.111

Simplify: 3x+25x23x24.3x+25x23x24.

Try It 8.112

Simplify: 2x71x+76x+71x249.2x71x+76x+71x249.

Example 8.57

Simplify: 4m27m+123m32m4.4m27m+123m32m4.

Try It 8.113

Simplify: 3x2+7x+104x+2+1x+5.3x2+7x+104x+2+1x+5.

Try It 8.114

Simplify: 4yy+5+2y+63yy2+11y+30.4yy+5+2y+63yy2+11y+30.

Example 8.58

Simplify: yy+11+1y1.yy+11+1y1.

Try It 8.115

Simplify: xx+31+1x+3.xx+31+1x+3.

Try It 8.116

Simplify: 1+1x13x+1.1+1x13x+1.

Section 8.5 Exercises

Practice Makes Perfect

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

255.

2aa+44a2a2162aa+44a2a216

256.

3bb5b2b2253bb5b2b225

257.

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

258.

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

259.

12+5623+7912+5623+79

260.

12+3435+71012+3435+710

261.

231934+56231934+56

262.

121623+34121623+34

263.

nm+1n1nnmnm+1n1nnm

264.

1p+pqqp1q1p+pqqp1q

265.

1r+1t1r21t21r+1t1r21t2

266.

2v+2w1v21w22v+2w1v21w2

267.

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

268.

y2yy42y42y+4y2yy42y42y+4

269.

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

270.

44b51b5+b444b51b5+b4

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

271.

13+1814+11213+1814+112

272.

14+1916+11214+1916+112

273.

56+297181356+2971813

274.

16+415351216+4153512

275.

cd+1d1ddccd+1d1ddc

276.

1m+mnnm1n1m+mnnm1n

277.

1p+1q1p21q21p+1q1p21q2

278.

2r+2t1r21t22r+2t1r21t2

279.

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

280.

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

281.

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

282.

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

283.

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

284.

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

285.

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

286.

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

287.

2+1p35p32+1p35p3

288.

nn23+5n2nn23+5n2

289.

mm+54+1m5mm+54+1m5

290.

7+2q21q+27+2q21q+2

Simplify

In the following exercises, use either method.

291.

342712+514342712+514

292.

vw+1v1vvwvw+1v1vvw

293.

2a+41a2162a+41a216

294.

3b23b405b+52b83b23b405b+52b8

295.

3m+3n1m21n23m+3n1m21n2

296.

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

297.

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

298.

yy+32+1y3yy+32+1y3

Everyday Math

299.

Electronics The resistance of a circuit formed by connecting two resistors in parallel is 11R1+1R211R1+1R2.

  1. Simplify the complex fraction 11R1+1R211R1+1R2.
  2. Find the resistance of the circuit when R1=8R1=8 and R2=12R2=12.
300.

Ironing Lenore can do the ironing for her family’s business in hh hours. Her daughter would take h+2h+2 hours to get the ironing done. If Lenore and her daughter work together, using 2 irons, the number of hours it would take them to do all the ironing is 11h+1h+211h+1h+2.

  1. Simplify the complex fraction 11h+1h+211h+1h+2.
  2. Find the number of hours it would take Lenore and her daughter, working together, to get the ironing done if h=4h=4.

Writing Exercises

301.

In this section, you learned to simplify the complex fraction 3x+2xx243x+2xx24 two ways:

rewriting it as a division problem

multiplying the numerator and denominator by the LCD

Which method do you prefer? Why?

302.

Efraim wants to start simplifying the complex fraction 1a+1b1a1b1a+1b1a1b by cancelling the variables from the numerator and denominator. 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.

The above image is four columns and three rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “simplify a complex rational expression by writing it as division.”, the next row reads “simplify a complex rational expression by using the LCD.” The remaining columns are blank.

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

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