Divide ((x^2+10x+16)/(x^2-6x-16))÷((x+8)/(x^2-64))

$\frac{x^{2} + 10 x + 16}{x^{2} - 6 x - 16} \div \frac{x + 8}{x^{2} - 64}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 10 x + 16}{x^{2} - 6 x - 16} \cdot \frac{x^{2} - 64}{x + 8}$

Step 2

Factor $x^{2} + 10 x + 16$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $16$ and whose sum is $10$ .

$2, 8$

Write the factored form using these integers.

$\frac{(x + 2) (x + 8)}{x^{2} - 6 x - 16} \cdot \frac{x^{2} - 64}{x + 8}$

Step 3

Factor $x^{2} - 6 x - 16$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 16$ and whose sum is $- 6$ .

$- 8, 2$

Write the factored form using these integers.

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

Step 4

Cancel the common factor of $x + 8$ .

Factor $x + 8$ out of $(x + 2) (x + 8)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 5

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{x - 8} (x^{2} - 64)$

Step 6

Multiply $\frac{1}{x - 8}$ by $x^{2} - 64$ .

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

Step 7

Simplify the numerator.

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

$\frac{x^{2} - 8^{2}}{x - 8}$

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 $b = 8$ .

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

Step 8

Cancel the common factor of $x - 8$ .

Cancel the common factor.

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

Divide $x + 8$ by $1$ .

$x + 8$

Divide ((x^2+10x+16)/(x^2-6x-16))÷((x+8)/(x^2-64))

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