Question

In the following exercises, perform the indicated operations

$\frac{5q}{p^{2}q-p^2}+\frac{4q}{q^2-1}$

Short answer

Simplified expression is $\frac{5q^2+5q+4p^{2}q}{p^2(q^2-1)}$

Step-by-step solution

Step 1. Given expression   

$\frac{5q}{p^{2}q-p^2}+\frac{4q}{q^2-1}$

Step 2. Find LCD    

Denominator of first fraction is $p^{2}q-p^2$

Factorize this denominator

$p^2(q-1)$

Denominator of the second fraction is$q^2-1$

Factorize this denominator

$q^2-1^2=(q-1)(q+!)$

here $(q-1)$is common between both denominator

So the LCD is$p^2(q-1)(q+1)$

Step 3. Rewrite each rational expression as  an equivalent rational expression with the LCD  

$\frac{5q(q+1)}{p^2(q-1)(q+1)}+\frac{4q.p^2}{p^2(q-1)(q+1)}$

Step 4.Simplify the numerators.     

$\frac{5q^2+5q}{p^2(q-1)(q+1)}+\frac{4p^{2}q}{p^2(q-1)(q+1)}$

Step 5. Add numerators of the fractions  

$$\begin{gathered} \frac{5q^2+5q+4p^{2}q}{p^2(q-1)(q+1)} \\ =\frac{5q^2+5q+4p^{2}q}{p^2(q^2-1)} \end{gathered}$$