Question

In the following exercises, simplify.

$\frac{{\displaystyle \frac{3}{s+6}}+{\displaystyle \frac{5}{s-6}}}{{\displaystyle \frac{1}{s^2-36}}+{\displaystyle \frac{4}{s+6}}}$

Short answer

The simplified expression is$\frac{8s+12}{4s-23}$

Step-by-step solution

Step 1. Given information 

The given expression is

$\frac{{\displaystyle \frac{3}{s+6}+\frac{5}{s-6}}}{{\displaystyle \frac{1}{s^2-36}+\frac{4}{s+6}}}$

Step 2. Simplify the expression 

$$\begin{gathered} \frac{{\displaystyle \frac{3}{s+6}+\frac{5}{s-6}}}{{\displaystyle \frac{1}{s^2-36}+\frac{4}{s+6}}}=\frac{\left(\cancel{s+6}\right)\left(\cancel{s-6}\right)\left({\displaystyle \frac{3}{\cancel{s+6}}}+{\displaystyle \frac{5}{\left(\cancel{s-6}\right)}}\right)}{\left(\cancel{s+6}\right)\left(\cancel{s-6}\right)\left({\displaystyle \frac{1}{\left(\cancel{s+6}\right)\left(\cancel{s-6}\right)}}+{\displaystyle \frac{4}{\cancel{s+6}}}\right)} \\ =\frac{3\left(s-6\right)+5\left(s+6\right)}{1+4\left(s-6\right)} \\ =\frac{8s+12}{4s-23} \end{gathered}$$

The simplified expression isrole="math" localid="1647792054607" $\frac{8s+12}{4s-23}$

Step 3. The solution

The simplified solution is$\frac{8s+12}{4s-23}$