### 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.

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

If you missed this problem, review Example 1.27.

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

If you missed this problem, review Example 1.31.

Solve: $\frac{1}{2x}+\frac{1}{4}=\frac{1}{8}.$

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:

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:

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.

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 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: $\frac{\frac{6}{x-4}}{\frac{3}{{x}^{2}-16}}.$

Simplify the complex rational expression by writing it as division: $\frac{\frac{2}{{x}^{2}-1}}{\frac{3}{x+1}}.$

Simplify the complex rational expression by writing it as division: $\frac{\frac{1}{{x}^{2}-7x+12}}{\frac{2}{x-4}}.$

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 7.25

Simplify the complex rational expression by writing it as division: $\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}.$

Simplify the complex rational expression by writing it as division: $\frac{\frac{1}{2}+\frac{2}{3}}{\frac{5}{6}+\frac{1}{12}}.$

Simplify the complex rational expression by writing it as division: $\frac{\frac{3}{4}-\frac{1}{3}}{\frac{1}{8}+\frac{5}{6}}.$

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

Simplify the complex rational expression by writing it as division: $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x}-\frac{1}{y}}.$

Simplify the complex rational expression by writing it as division: $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{{a}^{2}}-\frac{1}{{b}^{2}}}$.

We summarize the steps here.

### 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 7.27

Simplify the complex rational expression by writing it as division: $\frac{n-\frac{4n}{n+5}}{\frac{1}{n+5}+\frac{1}{n-5}}.$

Simplify the complex rational expression by writing it as division: $\frac{b-\frac{3b}{b+5}}{\frac{2}{b+5}+\frac{1}{b-5}}.$

Simplify the complex rational expression by writing it as division: $\frac{1-\frac{3}{c+4}}{\frac{1}{c+4}+\frac{c}{3}}.$

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

### Example 7.28

Simplify the complex rational expression by using the LCD: $\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}-\frac{1}{3}}.$

Simplify the complex rational expression by using the LCD: $\frac{\frac{1}{2}+\frac{1}{5}}{\frac{1}{10}+\frac{1}{5}}.$

Simplify the complex rational expression by using the LCD: $\frac{\frac{1}{4}+\frac{3}{8}}{\frac{1}{2}-\frac{5}{16}}.$

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

Simplify the complex rational expression by using the LCD: $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{a}{b}+\frac{b}{a}}.$

Simplify the complex rational expression by using the LCD: $\frac{\frac{1}{{x}^{2}}-\frac{1}{{y}^{2}}}{\frac{1}{x}+\frac{1}{y}}.$

### 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 7.30

Simplify the complex rational expression by using the LCD: $\frac{\frac{2}{x+6}}{\frac{4}{x-6}-\frac{4}{{x}^{2}-36}}.$

Simplify the complex rational expression by using the LCD: $\frac{\frac{3}{x+2}}{\frac{5}{x-2}-\frac{3}{{x}^{2}-4}}.$

Simplify the complex rational expression by using the LCD: $\frac{\frac{2}{x-7}-\frac{1}{x+7}}{\frac{6}{x+7}-\frac{1}{{x}^{2}-49}}.$

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

Simplify the complex rational expression by using the LCD: $\frac{\frac{3}{{x}^{2}+7x+10}}{\frac{4}{x+2}+\frac{1}{x+5}}.$

Simplify the complex rational expression by using the LCD: $\frac{\frac{4y}{y+5}+\frac{2}{y+6}}{\frac{3y}{{y}^{2}+11y+30}}.$

### Example 7.32

Simplify the complex rational expression by using the LCD: $\frac{\frac{y}{y+1}}{1+\frac{1}{y-1}}.$

Simplify the complex rational expression by using the LCD: $\frac{\frac{x}{x+3}}{1+\frac{1}{x+3}}.$

Simplify the complex rational expression by using the LCD: $\frac{1+\frac{1}{x-1}}{\frac{3}{x+1}}.$

### Media

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

#### 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 or 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, $\frac{\frac{1}{\overline{)a}}+\frac{1}{\overline{)b}}}{\frac{1}{\overline{)a}}-\frac{1}{\overline{)b}}}.$ 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?