### 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

Before you get started, take this readiness quiz.

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

Simplify: $\frac{\frac{3}{5}}{\frac{9}{10}}.$

If you missed this problem, review Example 1.72.

Simplify: $\frac{1-\frac{1}{3}}{{4}^{2}+4\xb75}.$

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:

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:

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.

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

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: $\frac{\frac{4}{y-3}}{\frac{8}{{y}^{2}-9}}.$

Simplify: $\frac{\frac{2}{{x}^{2}-1}}{\frac{3}{x+1}}.$

Simplify: $\frac{\frac{1}{{x}^{2}-7x+12}}{\frac{2}{x-4}}.$

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: $\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}.$

Simplify: $\frac{\frac{1}{2}+\frac{2}{3}}{\frac{5}{6}+\frac{1}{12}}.$

Simplify: $\frac{\frac{3}{4}-\frac{1}{3}}{\frac{1}{8}+\frac{5}{6}}.$

### Example 8.52

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

Simplify: $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}.$

Simplify: $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x}-\frac{1}{y}}.$

Simplify: $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{{a}^{2}}-\frac{1}{{b}^{2}}}.$

### How To

#### Simplify a complex rational expression by writing it as division.

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

### Example 8.53

Simplify: $\frac{n-\frac{4n}{n+5}}{\frac{1}{n+5}+\frac{1}{n-5}}.$

Simplify: $\frac{b-\frac{3b}{b+5}}{\frac{2}{b+5}+\frac{1}{b-5}}.$

Simplify: $\frac{1-\frac{3}{c+4}}{\frac{1}{c+4}+\frac{c}{3}}.$

### 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 $\frac{\text{LCD}}{\text{LCD}}$ we are multiplying by 1, so the value stays the same.

### Example 8.54

Simplify: $\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}.$

Simplify: $\frac{\frac{1}{2}+\frac{1}{5}}{\frac{1}{10}+\frac{1}{5}}.$

Simplify: $\frac{\frac{1}{4}+\frac{3}{8}}{\frac{1}{2}-\frac{5}{16}}.$

### Example 8.55

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

Simplify: $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}.$

Simplify: $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{a}{b}+\frac{b}{a}}.$

Simplify: $\frac{\frac{1}{{x}^{2}}-\frac{1}{{y}^{2}}}{\frac{1}{x}+\frac{1}{y}}.$

### How To

#### Simplify a complex rational expression by using the LCD.

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

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

### Example 8.56

Simplify: $\frac{\frac{2}{x+6}}{\frac{4}{x-6}-\frac{4}{{x}^{2}-36}}.$

Simplify: $\frac{\frac{3}{x+2}}{\frac{5}{x-2}-\frac{3}{{x}^{2}-4}}.$

Simplify: $\frac{\frac{2}{x-7}-\frac{1}{x+7}}{\frac{6}{x+7}-\frac{1}{{x}^{2}-49}}.$

### Example 8.57

Simplify: $\frac{\frac{4}{{m}^{2}-7m+12}}{\frac{3}{m-3}-\frac{2}{m-4}}.$

Simplify: $\frac{\frac{3}{{x}^{2}+7x+10}}{\frac{4}{x+2}+\frac{1}{x+5}}.$

Simplify: $\frac{\frac{4}{y+5}+\frac{2}{y+6}}{\frac{3y}{{y}^{2}+11y+30}}.$

### Example 8.58

Simplify: $\frac{\frac{y}{y+1}}{1+\frac{1}{y-1}}.$

Simplify: $\frac{\frac{x}{x+3}}{1+\frac{1}{x+3}}.$

Simplify: $\frac{1+\frac{1}{x-1}}{\frac{3}{x+1}}.$

### Section 8.5 Exercises

#### Practice Makes Perfect

**Simplify a Complex Rational Expression by Writing It as Division**

In the following exercises, simplify.

$\frac{\frac{3b}{b-5}}{\frac{{b}^{2}}{{b}^{2}-25}}$

$\frac{\frac{8}{{d}^{2}+9d+18}}{\frac{12}{d+6}}$

$\frac{\frac{1}{2}+\frac{3}{4}}{\frac{3}{5}+\frac{7}{10}}$

$\frac{\frac{1}{2}-\frac{1}{6}}{\frac{2}{3}+\frac{3}{4}}$

$\frac{\frac{1}{p}+\frac{p}{q}}{\frac{q}{p}-\frac{1}{q}}$

$\frac{\frac{2}{v}+\frac{2}{w}}{\frac{1}{{v}^{2}}-\frac{1}{{w}^{2}}}$

$\frac{y-\frac{2y}{y-4}}{\frac{2}{y-4}-\frac{2}{y+4}}$

$\frac{4-\frac{4}{b-5}}{\frac{1}{b-5}+\frac{b}{4}}$

**Simplify a Complex Rational Expression by Using the LCD**

In the following exercises, simplify.

$\frac{\frac{1}{4}+\frac{1}{9}}{\frac{1}{6}+\frac{1}{12}}$

$\frac{\frac{1}{6}+\frac{4}{15}}{\frac{3}{5}-\frac{1}{2}}$

$\frac{\frac{1}{m}+\frac{m}{n}}{\frac{n}{m}-\frac{1}{n}}$

$\frac{\frac{2}{r}+\frac{2}{t}}{\frac{1}{{r}^{2}}-\frac{1}{{t}^{2}}}$

$\frac{\frac{5}{y-4}}{\frac{3}{y+4}+\frac{2}{{y}^{2}-16}}$

$\frac{\frac{3}{s+6}+\frac{5}{s-6}}{\frac{1}{{s}^{2}-36}+\frac{4}{s+6}}$

$\frac{\frac{5}{{b}^{2}-6b-27}}{\frac{3}{b-9}+\frac{1}{b+3}}$

$\frac{\frac{6}{d-4}-\frac{2}{d+7}}{\frac{2d}{{d}^{2}+3d-28}}$

$\frac{\frac{n}{n-2}}{3+\frac{5}{n-2}}$

$\frac{7+\frac{2}{q-2}}{\frac{1}{q+2}}$

**Simplify**

In the following exercises, use either method.

$\frac{\frac{v}{w}+\frac{1}{v}}{\frac{1}{v}-\frac{v}{w}}$

$\frac{\frac{3}{{b}^{2}-3b-40}}{\frac{5}{b+5}-\frac{2}{b-8}}$

$\frac{\frac{2}{r-9}}{\frac{1}{r+9}+\frac{3}{{r}^{2}-81}}$

$\frac{\frac{y}{y+3}}{2+\frac{1}{y-3}}$

#### Everyday Math

**Electronics** The resistance of a circuit formed by connecting two resistors in parallel is $\frac{1}{\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}}$.

- ⓐ Simplify the complex fraction $\frac{1}{\frac{1}{{R}_{1}}+\frac{1}{{R}_{2}}}$.
- ⓑ Find the resistance of the circuit when ${R}_{1}=8$ and ${R}_{2}=12$.

**Ironing** Lenore can do the ironing for her family’s business in $h$ hours. Her daughter would take $h+2$ 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 $\frac{1}{\frac{1}{h}+\frac{1}{h+2}}$.

- ⓐ Simplify the complex fraction $\frac{1}{\frac{1}{h}+\frac{1}{h+2}}$.
- ⓑ Find the number of hours it would take Lenore and her daughter, working together, to get the ironing done if $h=4$.

#### Writing Exercises

In this section, you learned to simplify the complex fraction $\frac{\frac{3}{x+2}}{\frac{x}{{x}^{2}-4}}$ two ways:

rewriting it as a division problem

multiplying the numerator and denominator by the LCD

Which method do you prefer? Why?

Efraim wants to start simplifying the complex fraction $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{a}-\frac{1}{b}}$ 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?