Question

In the following exercises, simplify each complex rational expression using either method.

$\frac{{\displaystyle \frac{3}{b^2-3b-40}}}{{\displaystyle \frac{5}{b+5}}-{\displaystyle \frac{2}{b-8}}}$

Short answer

The simplified form is $\frac{3}{3b-50}$

Step-by-step solution

Step 1. Given. 

The given complex rational expression is: $\frac{{\displaystyle \frac{3}{b^2-3b-40}}}{{\displaystyle \frac{5}{b+5}-\frac{2}{b-8}}}$.

Step 2. To find 

We have to simplify each complex rational expression by using either method.

Step 3. Calculation  

LCM of the denominators is = $\left(b+5\right)\left(b-8\right)$. Multiply numerator and denominator by it to simplify.

$$\begin{gathered} \frac{{\displaystyle \frac{3}{b^2-3b-40}}}{{\displaystyle \frac{5}{b+5}-\frac{2}{b-8}}} \\ =\frac{{\displaystyle \frac{3}{b^2-3b-40}\left(b+5\right)\left(b-8\right)}}{{\displaystyle \frac{5}{b+5}\left(b+5\right)\left(b-8\right)-\frac{2}{b-8}\left(b+5\right)\left(b-8\right)}} \\ =\frac{3}{5(b-8)-2(b+5)} \\ =\frac{3}{5b-40-2b-10} \\ =\frac{3}{3b-50} \end{gathered}$$