8.5 Simplify complex rational expressions

Solution

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

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 $y=\mathrm{-3}$ .

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.

Solution