Simplify 7/(a+8)+7/(a^2-64)

$\frac{7}{a + 8} + \frac{7}{a^{2} - 64}$

Step 1

Simplify the denominator.

Rewrite $64$ as $8^{2}$ .

$\frac{7}{a + 8} + \frac{7}{a^{2} - 8^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = a$ and $b = 8$ .

$\frac{7}{a + 8} + \frac{7}{(a + 8) (a - 8)}$

Step 2

To write $\frac{7}{a + 8}$ as a fraction with a common denominator, multiply by $\frac{a - 8}{a - 8}$ .

$\frac{7}{a + 8} \cdot \frac{a - 8}{a - 8} + \frac{7}{(a + 8) (a - 8)}$

Step 3

Simplify terms.

Multiply $\frac{7}{a + 8}$ by $\frac{a - 8}{a - 8}$ .

$\frac{7 (a - 8)}{(a + 8) (a - 8)} + \frac{7}{(a + 8) (a - 8)}$

Combine the numerators over the common denominator.

$\frac{7 (a - 8) + 7}{(a + 8) (a - 8)}$

Step 4

Simplify the numerator.

Factor $7$ out of $7 (a - 8) + 7$ .

Factor $7$ out of $7$ .

$\frac{7 (a - 8) + 7 (1)}{(a + 8) (a - 8)}$

Factor $7$ out of $7 (a - 8) + 7 (1)$ .

$\frac{7 (a - 8 + 1)}{(a + 8) (a - 8)}$

Add $- 8$ and $1$ .

$\frac{7 (a - 7)}{(a + 8) (a - 8)}$