EDGE Math 8

Complex Rational Expressions

Definition

A rational expression that has one or more rational expressions in its numerator, denominator, or both is called a complex rational expression.

Below are some examples of complex rational expressions.

\frac{\frac{1}{2}}{x}

\frac{3 x}{\frac{2}{y}}

\frac{1 + \frac{1}{x}}{\frac{2}{3} + \frac{3}{x}}

Recall that a fraction indicates division. Hence, a complex rational expression can be expressed in simplest form by simplifying its numerator and denominator first. Then express the result as division.

Example 1

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

Solution

\frac{\frac{x - 1}{x + 1}}{\frac{x + 2}{x^{2} - 1}}

= \frac{x - 1}{x + 1} \div \frac{x + 2}{x^{2} - 1}

= \frac{x - 1}{x + 1} \cdot \frac{x^{2} - 1}{x + 2}

= \frac{(x - 1) (x^{2} - 1)}{(x + 1) (x + 2)}

= \frac{(x - 1) (x + 1) (x - 1)}{(x + 1) (x + 2)}

= \frac{(x - 1) (x - 1)}{x + 2} or \frac{{(x - 1)}^{2}}{x + 2}