8.5 Simplify complex rational expressions

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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.

1. Simplify: $\frac{\frac{3}{5}}{\frac{9}{10}}.$
If you missed this problem, review [link] .
2. Simplify: $\frac{1-\frac{1}{3}}{{4}^{2}+4·5}.$
If you missed this problem, review [link] .

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

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

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:

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

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

We will use two methods to simplify complex rational expressions.

Simplify a complex rational expression by writing it as division

We have already seen this complex rational expression earlier in this chapter.

$\frac{\frac{6{x}^{2}-7x+2}{4x-8}}{\frac{2{x}^{2}-8x+3}{{x}^{2}-5x+6}}$

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

$\left(\frac{6{x}^{2}-7x+2}{4x-8}\right)÷\left(\frac{2{x}^{2}-8x+3}{{x}^{2}-5x+6}\right)$

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.

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

Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{\frac{4}{y-3}}{\frac{8}{{y}^{2}-9}}\hfill \\ \\ \\ \text{Rewrite the complex fraction as division.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4}{y-3}÷\frac{8}{{y}^{2}-9}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite as the product of first times the}\hfill \\ \text{reciprocal of the second.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4}{y-3}·\frac{{y}^{2}-9}{8}\hfill \\ \\ \\ \text{Multiply.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left({y}^{2}-9\right)}{8\left(y-3\right)}\hfill \\ \\ \\ \text{Factor to look for common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left(y-3\right)\left(y+3\right)}{4·2\left(y-3\right)}\hfill \\ \\ \\ \text{Remove common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\overline{)4}\overline{)\left(y-3\right)}\left(y+3\right)}{\overline{)4}·2\overline{)\left(y-3\right)}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{y+3}{2}\hfill \end{array}$

Are there any value(s) of $y$ that should not be allowed? The simplified rational expression has just a constant in the denominator. But the original complex rational expression    had denominators of $y-3$ and ${y}^{2}-9$ . This expression would be undefined if $y=3$ or $y=-3$ .

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

$\frac{2}{3\left(x-1\right)}$

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

$\frac{1}{2\left(x-3\right)}$

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.

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

Solution

 Simplify the numerator and denominator. Find the LCD and add the fractions in the numerator. Find the LCD and add the fractions in the denominator. Simplify the numerator and denominator. Simplify the numerator and denominator, again. Rewrite the complex rational expression as a division problem. Multiply the first times by the reciprocal of the second. Simplify.

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

$\frac{14}{11}$

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

$\frac{10}{23}$

How to simplify a complex rational expression by writing it as division

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

Solution

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

$\frac{y+x}{y-x}$

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

$\frac{ab}{b-a}$

Simplify a complex rational expression by writing it as division.

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

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