Question

In the following exercises, simplify:

$\frac{{\displaystyle \frac{5}{y-4}}}{{\displaystyle \frac{3}{y+4}}+{\displaystyle \frac{2}{y^2-16}}}$.

Short answer

The simplified rational expression is,$\frac{5y+20}{3y-10}$.

Step-by-step solution

Step 1. Given Information.

The rational expression is,

$\frac{{\displaystyle \frac{5}{y-4}}}{{\displaystyle \frac{3}{y+4}}+{\displaystyle \frac{2}{y^2-16}}}$.

Step 2. Simplifying the rational expression.

$\frac{{\displaystyle \frac{5}{y-4}}}{{\displaystyle \frac{3}{y+4}}+{\displaystyle \frac{2}{y^2-16}}}$

The LCD of the expression is $(y+4)(y-4)$

$$\begin{gathered} =\frac{(y-4)(y+4)\left({\displaystyle \frac{5}{y-4}}\right)}{(y-4)(y+4)({\displaystyle \frac{3}{y+4}}+{\displaystyle \frac{2}{y^2-16}})} \\ =\frac{(y-4)(y+4)\left({\displaystyle \frac{5}{y-4}}\right)}{(y-4)(y+4)\left({\displaystyle \frac{3}{y+4}}\right)+(y-4)(y+4)\left({\displaystyle \frac{2}{(y-4)(y+4)}}\right)} \\ =\frac{5(y+4)}{3(y-4)+2} \\ =\frac{5y+20}{3y-12+2} \\ =\frac{5y+20}{3y-10} \end{gathered}$$