Elementary Algebra 2e

# 8.5Simplify Complex Rational Expressions

Elementary Algebra 2e8.5 Simplify Complex Rational Expressions

### 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 8.17

Before you get started, take this readiness quiz.

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

Simplify: $35910.35910.$
If you missed this problem, review Example 1.72.

Be Prepared 8.18

Simplify: $1−1342+4·5.1−1342+4·5.$
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:

$4y−38y2−91x+1yxy−yx2x+64x−6−4x2−364y−38y2−91x+1yxy−yx2x+64x−6−4x2−36$

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.

$6x2−7x+24x−82x2−8x+3x2−5x+66x2−7x+24x−82x2−8x+3x2−5x+6$

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

$(6x2−7x+24x−8)÷(2x2−8x+3x2−5x+6)(6x2−7x+24x−8)÷(2x2−8x+3x2−5x+6)$

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: $4y−38y2−9.4y−38y2−9.$

Try It 8.99

Simplify: $2x2−13x+1.2x2−13x+1.$

Try It 8.100

Simplify: $1x2−7x+122x−4.1x2−7x+122x−4.$

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: $13+1612−13.13+1612−13.$

Try It 8.101

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

Try It 8.102

Simplify: $34−1318+56.34−1318+56.$

### Example 8.52

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

Simplify: $1x+1yxy−yx.1x+1yxy−yx.$

Try It 8.103

Simplify: $1x+1y1x−1y.1x+1y1x−1y.$

Try It 8.104

Simplify: $1a+1b1a2−1b2.1a+1b1a2−1b2.$

### 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: $n−4nn+51n+5+1n−5.n−4nn+51n+5+1n−5.$

Try It 8.105

Simplify: $b−3bb+52b+5+1b−5.b−3bb+52b+5+1b−5.$

Try It 8.106

Simplify: $1−3c+41c+4+c3.1−3c+41c+4+c3.$

### 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: $13+1612−13.13+1612−13.$

Try It 8.107

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

Try It 8.108

Simplify: $14+3812−516.14+3812−516.$

### Example 8.55

#### How to Simplify a Complex Rational Expression by Using the LCD

Simplify: $1x+1yxy−yx.1x+1yxy−yx.$

Try It 8.109

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

Try It 8.110

Simplify: $1x2−1y21x+1y.1x2−1y21x+1y.$

### 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: $2x+64x−6−4x2−36.2x+64x−6−4x2−36.$

Try It 8.111

Simplify: $3x+25x−2−3x2−4.3x+25x−2−3x2−4.$

Try It 8.112

Simplify: $2x−7−1x+76x+7−1x2−49.2x−7−1x+76x+7−1x2−49.$

### Example 8.57

Simplify: $4m2−7m+123m−3−2m−4.4m2−7m+123m−3−2m−4.$

Try It 8.113

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

Try It 8.114

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

### Example 8.58

Simplify: $yy+11+1y−1.yy+11+1y−1.$

Try It 8.115

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

Try It 8.116

Simplify: $1+1x−13x+1.1+1x−13x+1.$

### Section 8.5 Exercises

#### Practice Makes Perfect

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

255.

$2aa+44a2a2−162aa+44a2a2−16$

256.

$3bb−5b2b2−253bb−5b2b2−25$

257.

$5c2+5c−1410c+75c2+5c−1410c+7$

258.

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

259.

$12+5623+7912+5623+79$

260.

$12+3435+71012+3435+710$

261.

$23−1934+5623−1934+56$

262.

$12−1623+3412−1623+34$

263.

$nm+1n1n−nmnm+1n1n−nm$

264.

$1p+pqqp−1q1p+pqqp−1q$

265.

$1r+1t1r2−1t21r+1t1r2−1t2$

266.

$2v+2w1v2−1w22v+2w1v2−1w2$

267.

$x−2xx+31x+3+1x−3x−2xx+31x+3+1x−3$

268.

$y−2yy−42y−4−2y+4y−2yy−42y−4−2y+4$

269.

$2−2a+31a+3+a22−2a+31a+3+a2$

270.

$4−4b−51b−5+b44−4b−51b−5+b4$

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.

$56+29718−1356+29718−13$

274.

$16+41535−1216+41535−12$

275.

$cd+1d1d−dccd+1d1d−dc$

276.

$1m+mnnm−1n1m+mnnm−1n$

277.

$1p+1q1p2−1q21p+1q1p2−1q2$

278.

$2r+2t1r2−1t22r+2t1r2−1t2$

279.

$2x+53x−5+1x2−252x+53x−5+1x2−25$

280.

$5y−43y+4+2y2−165y−43y+4+2y2−16$

281.

$5z2−64+3z+81z+8+2z−85z2−64+3z+81z+8+2z−8$

282.

$3s+6+5s−61s2−36+4s+63s+6+5s−61s2−36+4s+6$

283.

$4a2−2a−151a−5+2a+34a2−2a−151a−5+2a+3$

284.

$5b2−6b−273b−9+1b+35b2−6b−273b−9+1b+3$

285.

$5c+2−3c+75cc2+9c+145c+2−3c+75cc2+9c+14$

286.

$6d−4−2d+72dd2+3d−286d−4−2d+72dd2+3d−28$

287.

$2+1p−35p−32+1p−35p−3$

288.

$nn−23+5n−2nn−23+5n−2$

289.

$mm+54+1m−5mm+54+1m−5$

290.

$7+2q−21q+27+2q−21q+2$

Simplify

In the following exercises, use either method.

291.

$34−2712+51434−2712+514$

292.

$vw+1v1v−vwvw+1v1v−vw$

293.

$2a+41a2−162a+41a2−16$

294.

$3b2−3b−405b+5−2b−83b2−3b−405b+5−2b−8$

295.

$3m+3n1m2−1n23m+3n1m2−1n2$

296.

$2r−91r+9+3r2−812r−91r+9+3r2−81$

297.

$x−3xx+23x+2+3x−2x−3xx+23x+2+3x−2$

298.

$yy+32+1y−3yy+32+1y−3$

#### 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 $3x+2xx2−43x+2xx2−4$ 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 $1a+1b1a−1b1a+1b1a−1b$ 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. After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

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