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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.
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.
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.
Simplify: $\frac{\frac{4}{y3}}{\frac{8}{{y}^{2}9}}.$
$\begin{array}{cccc}& & & \hfill \phantom{\rule{5em}{0ex}}\frac{\frac{4}{y3}}{\frac{8}{{y}^{2}9}}\hfill \\ \\ \\ \text{Rewrite the complex fraction as division.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4}{y3}\xf7\frac{8}{{y}^{2}9}\hfill \\ \\ \\ \begin{array}{c}\text{Rewrite as the product of first times the}\hfill \\ \text{reciprocal of the second.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4}{y3}\xb7\frac{{y}^{2}9}{8}\hfill \\ \\ \\ \text{Multiply.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left({y}^{2}9\right)}{8\left(y3\right)}\hfill \\ \\ \\ \text{Factor to look for common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{4\left(y3\right)\left(y+3\right)}{4\xb72\left(y3\right)}\hfill \\ \\ \\ \text{Remove common factors.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{\overline{)4}\overline{)\left(y3\right)}\left(y+3\right)}{\overline{)4}\xb72\overline{)\left(y3\right)}}\hfill \\ \\ \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{5em}{0ex}}\frac{y+3}{2}\hfill \end{array}$
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 $y3$ and ${y}^{2}9$ . This expression would be undefined if $y=3$ or $y=\mathrm{3}$ .
Simplify: $\frac{\frac{2}{{x}^{2}1}}{\frac{3}{x+1}}.$
$\frac{2}{3(x1)}$
Simplify: $\frac{\frac{1}{{x}^{2}7x+12}}{\frac{2}{x4}}.$
$\frac{1}{2(x3)}$
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: $\frac{\frac{1}{3}+\frac{1}{6}}{\frac{1}{2}\frac{1}{3}}.$
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: $\frac{\frac{3}{4}\frac{1}{3}}{\frac{1}{8}+\frac{5}{6}}.$
$\frac{10}{23}$
Simplify: $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{x}{y}\frac{y}{x}}.$
Simplify: $\frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x}\frac{1}{y}}.$
$\frac{y+x}{yx}$
Simplify: $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{{a}^{2}}\frac{1}{{b}^{2}}}.$
$\frac{ab}{ba}$
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