Simplify ((9x)/(x-8)*(x+4)/(x^2+12x+32))÷((3x)/(9x^3-576x))

$\frac{9 x}{x - 8} \cdot \frac{x + 4}{x^{2} + 12 x + 32} \div \frac{3 x}{9 x^{3} - 576 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{9 x}{x - 8} \cdot \frac{x + 4}{x^{2} + 12 x + 32} \frac{9 x^{3} - 576 x}{3 x}$

Step 2

Factor $x^{2} + 12 x + 32$ 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 $32$ and whose sum is $12$ .

$4, 8$

Write the factored form using these integers.

$\frac{9 x}{x - 8} \cdot \frac{x + 4}{(x + 4) (x + 8)} \frac{9 x^{3} - 576 x}{3 x}$

Step 3

Simplify terms.

Cancel the common factor of $x + 4$ .

Cancel the common factor.

$\frac{9 x}{x - 8} \cdot \frac{x + 4}{(x + 4) (x + 8)} \frac{9 x^{3} - 576 x}{3 x}$

Rewrite the expression.

$\frac{9 x}{x - 8} \cdot \frac{1}{x + 8} \frac{9 x^{3} - 576 x}{3 x}$

Multiply $\frac{9 x}{x - 8}$ by $\frac{1}{x + 8}$ .

$\frac{9 x}{(x - 8) (x + 8)} \cdot \frac{9 x^{3} - 576 x}{3 x}$

Cancel the common factor of $3 x$ .

Factor $3 x$ out of $9 x$ .

$\frac{3 x (3)}{(x - 8) (x + 8)} \cdot \frac{9 x^{3} - 576 x}{3 x}$

Cancel the common factor.

$\frac{3 x \cdot 3}{(x - 8) (x + 8)} \cdot \frac{9 x^{3} - 576 x}{3 x}$

Rewrite the expression.

$\frac{3}{(x - 8) (x + 8)} (9 x^{3} - 576 x)$

Multiply $\frac{3}{(x - 8) (x + 8)}$ by $9 x^{3} - 576 x$ .

$\frac{3 (9 x^{3} - 576 x)}{(x - 8) (x + 8)}$

Step 4

Simplify the numerator.

Factor $9 x$ out of $9 x^{3} - 576 x$ .

Factor $9 x$ out of $9 x^{3}$ .

$\frac{3 (9 x (x^{2}) - 576 x)}{(x - 8) (x + 8)}$

Factor $9 x$ out of $- 576 x$ .

$\frac{3 (9 x (x^{2}) + 9 x (- 64))}{(x - 8) (x + 8)}$

Factor $9 x$ out of $9 x (x^{2}) + 9 x (- 64)$ .

$\frac{3 (9 x (x^{2} - 64))}{(x - 8) (x + 8)}$

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

$\frac{3 (9 x (x^{2} - 8^{2}))}{(x - 8) (x + 8)}$

Factor.

$\frac{3 \cdot 9 x (x + 8) (x - 8)}{(x - 8) (x + 8)}$

Multiply $3$ by $9$ .

$\frac{27 x (x + 8) (x - 8)}{(x - 8) (x + 8)}$

Step 5

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x + 8$ .

Cancel the common factor.

$\frac{27 x (x + 8) (x - 8)}{(x - 8) (x + 8)}$

Rewrite the expression.

$\frac{27 x (x - 8)}{x - 8}$

Cancel the common factor of $x - 8$ .

Cancel the common factor.

$\frac{27 x (x - 8)}{x - 8}$

Divide $27 x$ by $1$ .

$27 x$

Simplify ((9x)/(x-8)*(x+4)/(x^2+12x+32))÷((3x)/(9x^3-576x))

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