Question

Simplify a Complex Rational Expression by Using the LCD.

In the following exercises, simplify each complex rational expression by using the LCD.$\frac{{\displaystyle \frac{3}{s+6}+\frac{5}{s-6}}}{{\displaystyle \frac{1}{s^2-36}+\frac{4}{s+6}}}$

Short answer

The simplification is $\frac{8s+12}{4s-23}$.

Step-by-step solution

Step 1. Given.

The given complex rational expression is: $\frac{{\displaystyle \frac{3}{s+6}+\frac{5}{s-6}}}{{\displaystyle \frac{1}{s^2-36}+\frac{4}{s+6}}}$.

Step 2. To find 

We have to simplify each complex rational expression by using the LCD.

Step 3. Calculation 

LCM of the denominators is =$\left(s+6\right)\left(s-6\right)$. Multiply numerator and denominator by it to simplify.

$$\begin{gathered} \frac{{\displaystyle \frac{3}{s+6}+\frac{5}{s-6}}}{{\displaystyle \frac{1}{s^2-36}+\frac{4}{s+6}}} \\ =\frac{{\displaystyle \frac{3}{s+6}\left(s+6\right)\left(s-6\right)+\frac{5}{s-6}\left(s+6\right)\left(s-6\right)}}{{\displaystyle \frac{1}{s^2-36}\left(s+6\right)\left(s-6\right)+\frac{4}{s+6}\left(s+6\right)\left(s-6\right)}} \\ =\frac{3(s-6)+5(s+6)}{1+4(s-6)} \\ =\frac{3s-18+5s+30}{1+4s-24} \\ =\frac{8s+12}{4s-23} \end{gathered}$$