Simplify (x^2+16x+64)/(x^2-64)

$\frac{x^{2} + 16 x + 64}{x^{2} - 64}$

Step 1

Factor using the perfect square rule.

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

$\frac{x^{2} + 16 x + 8^{2}}{x^{2} - 64}$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 x = 2 \cdot x \cdot 8$

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot 8 + 8^{2}}{x^{2} - 64}$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 8$ .

$\frac{{(x + 8)}^{2}}{x^{2} - 64}$

Step 2

Simplify the denominator.

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

$\frac{{(x + 8)}^{2}}{x^{2} - 8^{2}}$

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

$\frac{{(x + 8)}^{2}}{(x + 8) (x - 8)}$

Step 3

Cancel the common factor of ${(x + 8)}^{2}$ and $x + 8$ .

Factor $x + 8$ out of ${(x + 8)}^{2}$ .

$\frac{(x + 8) (x + 8)}{(x + 8) (x - 8)}$

Cancel the common factors.

Cancel the common factor.

$\frac{(x + 8) (x + 8)}{(x + 8) (x - 8)}$

Rewrite the expression.

$\frac{x + 8}{x - 8}$