Pre-Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

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

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Step 2.1

Factor $9$ out of $9 x$ .

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

Step 2.2

Factor $9$ out of $- 81$ .

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

Step 2.3

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.

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Step 3.1

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

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

Step 3.2

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.

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Step 4.1

Cancel the common factor of $x + 9$ .

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x - 9$ .

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Step 4.2.1

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

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.3

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

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

Step 4.4

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

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