Pre-Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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 $b = 9$ .

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

Step 3

Simplify terms.

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Step 3.1

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

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Step 3.1.1

Factor $9$ out of $9 x$ .

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

Step 3.1.2

Factor $9$ out of $81$ .

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

Step 3.1.3

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

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

Step 3.2

Cancel the common factor of $x + 9$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

$\frac{x - 9}{x} \cdot \frac{x - 9}{9}$

Step 3.3

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

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

Step 4

Raise $x - 9$ to the power of $1$ .

$\frac{{(x - 9)}^{1} (x - 9)}{x \cdot 9}$

Step 5

Raise $x - 9$ to the power of $1$ .

$\frac{{(x - 9)}^{1} {(x - 9)}^{1}}{x \cdot 9}$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{{(x - 9)}^{1 + 1}}{x \cdot 9}$

Step 7

Add $1$ and $1$ .

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

Step 8

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

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