Simplify (((x+9)^2)/(x-9))/((x^2-81)/(9x-81))

$\frac{\frac{{(x + 9)}^{2}}{x - 9}}{\frac{x^{2} - 81}{9 x - 81}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{{(x + 9)}^{2}}{x - 9} \cdot \frac{9 x - 81}{x^{2} - 81}$

Step 2

Factor $9$ out of $9 x - 81$ .

Factor $9$ out of $9 x$ .

$\frac{{(x + 9)}^{2}}{x - 9} \cdot \frac{9 (x) - 81}{x^{2} - 81}$

Factor $9$ out of $- 81$ .

$\frac{{(x + 9)}^{2}}{x - 9} \cdot \frac{9 x + 9 \cdot - 9}{x^{2} - 81}$

Factor $9$ out of $9 x + 9 \cdot - 9$ .

$\frac{{(x + 9)}^{2}}{x - 9} \cdot \frac{9 (x - 9)}{x^{2} - 81}$

Step 3

Simplify the denominator.

Rewrite $81$ as $9^{2}$ .

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

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 $b = 9$ .

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

Step 4

Simplify terms.

Cancel the common factor of $x + 9$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 9$ .

Factor $x - 9$ out of $9 (x - 9)$ .

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

Cancel the common factor.

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

Rewrite the expression.

$(x + 9) \frac{9}{x - 9}$

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

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

Move $9$ to the left of $x + 9$ .

$\frac{9 (x + 9)}{x - 9}$