Question

In the following exercises, perform the indicated operations

$\frac{8a}{a^2-64}-\frac{4}{a+8}$

Short answer

The simplified expression is $\frac{4}{a-8}$

Step-by-step solution

Step 1. Given information

Expression:

$\frac{8a}{a^2-64}-\frac{4}{a+8}$

Step 2. Find LCD      

Denominator of the first fraction is$a^2-64$

It can be factorize as

$a^2-8^2=(a-8)(a+8)$

Denominator of the second fraction is $a+8$

Here $\left(a+8\right)$is common to the both denominators

So LCD is$(a-8)(a+8)$

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

$\frac{8a}{(a-8)(a+8)}-\frac{4(a-8)}{(a+8)(a-8)}$

Step 4. Simplify numerators

$\frac{8a}{(a-8)(a+8)}-\frac{4a-32}{(a+8)(a-8)}$

Step 5. Subtract numerators

$$\begin{gathered} \frac{8a-(4a-32)}{(a-8)(a+8)} \\ =\frac{8a-4a+32}{(a-8)(a+8)} \\ =\frac{4a+32}{(a-8)(a+8)} \end{gathered}$$

Step 6. Take 4 common out of numerator and then cancel common factors

$$\begin{gathered} \frac{4(a+8)}{(a-8)(a+8)} \\ =\frac{4}{a-8} \end{gathered}$$