Mathematics » Rational Expressions and Equations » Simplify Complex Rational Expressions

# Defining Complex Rational Expressions

## What are complex rational expressions?

Complex fractions are fractions in which the numerator or denominator contains a fraction. In Chapter 1 we simplified complex fractions like these:

$$\cfrac{\frac{3}{4}}{\frac{5}{8}}\phantom{\rule{4em}{0ex}}\cfrac{\frac{x}{2}}{\frac{xy}{6}}$$

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:

$$\cfrac{\frac{4}{y-3}}{\frac{8}{{y}^{2}-9}}\phantom{\rule{7em}{0ex}}\cfrac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}-\frac{y}{x}}\phantom{\rule{7em}{0ex}}\cfrac{\frac{2}{x+6}}{\frac{4}{x-6}-\frac{4}{{x}^{2}-36}}$$

Remember, we always exclude values that would make any denominator zero.

We will use two methods to simplify complex rational expressions.