# 8.5 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 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.

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.

### 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.

### 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 LCDLCD we are multiplying by 1, so the value stays the same.

### 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.

### Section 8.5 Exercises

#### Practice Makes Perfect

Simplify a Complex Rational Expression by Writing It as Division

In the following exercises, simplify.

Simplify a Complex Rational Expression by Using the LCD

In the following exercises, simplify.

Simplify

In the following exercises, use either method.

Everyday Math

Writing Exercises

#### 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?