Simplify ((c^2-6c-16)/(c^2-d^2))÷((c^2-8c)/(c+d))

$\frac{c^{2} - 6 c - 16}{c^{2} - d^{2}} \div \frac{c^{2} - 8 c}{c + d}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{c^{2} - 6 c - 16}{c^{2} - d^{2}} \cdot \frac{c + d}{c^{2} - 8 c}$

Step 2

Factor $c^{2} - 6 c - 16$ 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.

$\frac{(c - 8) (c + 2)}{c^{2} - d^{2}} \cdot \frac{c + d}{c^{2} - 8 c}$

Step 3

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

$\frac{(c - 8) (c + 2)}{(c + d) (c - d)} \cdot \frac{c + d}{c^{2} - 8 c}$

Step 4

Simplify terms.

Cancel the common factor of $c + d$ .

Cancel the common factor.

$\frac{(c - 8) (c + 2)}{(c + d) (c - d)} \cdot \frac{c + d}{c^{2} - 8 c}$

Rewrite the expression.

$\frac{(c - 8) (c + 2)}{c - d} \cdot \frac{1}{c^{2} - 8 c}$

Multiply $\frac{(c - 8) (c + 2)}{c - d}$ by $\frac{1}{c^{2} - 8 c}$ .

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

Step 5

Factor $c$ out of $c^{2} - 8 c$ .

Factor $c$ out of $c^{2}$ .

$\frac{(c - 8) (c + 2)}{(c - d) (c \cdot c - 8 c)}$

Factor $c$ out of $- 8 c$ .

$\frac{(c - 8) (c + 2)}{(c - d) (c \cdot c + c \cdot - 8)}$

Factor $c$ out of $c \cdot c + c \cdot - 8$ .

$\frac{(c - 8) (c + 2)}{(c - d) (c (c - 8))}$

$\frac{(c - 8) (c + 2)}{(c - d) c (c - 8)}$

Step 6

Cancel the common factor of $c - 8$ .

Cancel the common factor.

$\frac{(c - 8) (c + 2)}{(c - d) c (c - 8)}$

Rewrite the expression.

$\frac{c + 2}{(c - d) c}$

Step 7

Reorder factors in $\frac{c + 2}{(c - d) c}$ .

$\frac{c + 2}{c (c - d)}$

Simplify ((c^2-6c-16)/(c^2-d^2))÷((c^2-8c)/(c+d))

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