Question

In the following exercises, simplify.

$\frac{{\displaystyle \frac{4}{a^2-2a-15}}}{{\displaystyle \frac{1}{a-5}}+{\displaystyle \frac{2}{a+3}}}$

Short answer

The simplified expression is$\frac{4}{3a-5}$

Step-by-step solution

Step 1. Given information 

The given expression is$\frac{{\displaystyle \frac{4}{a^2-2a-15}}}{{\displaystyle \frac{1}{a-5}+\frac{2}{a+3}}}$

Step2. Simplify The expression

$$\begin{gathered} \frac{{\displaystyle \frac{4}{a^2-2a-15}}}{{\displaystyle \frac{1}{a-5}+\frac{2}{a+3}}}=\frac{\left(\cancel{a+3}\right)\left(\cancel{a-5}\right)\left({\displaystyle \frac{4}{\left(\cancel{a+3}\right)\left(\cancel{a-5}\right)}}\right)}{\left(\cancel{a+3}\right)\left(\cancel{a-5}\right)\left({\displaystyle \frac{1}{\left(\cancel{a-5}\right)}}+{\displaystyle \frac{2}{\left(\cancel{a+3}\right)}}\right)} \\ =\frac{4}{\left(a+5\right)+2\left(a-5\right)} \\ =\frac{4}{a+5+2a-10} \\ =\frac{4}{3a-5} \end{gathered}$$

The simplified expression is$\frac{4}{3a-5}$

Step 3. The solution 

The simplified expression is$\frac{4}{3a-5}$