Simplify (1/9-1/x)/(1/81-1/(x^2))

$\frac{\frac{1}{9} - \frac{1}{x}}{\frac{1}{81} - \frac{1}{x^{2}}}$

Multiply the numerator and denominator of the complex fraction by $81 x^{2}$ .

Multiply $\frac{\frac{1}{9} - \frac{1}{x}}{\frac{1}{81} - \frac{1}{x^{2}}}$ by $\frac{81 x^{2}}{81 x^{2}}$ .

$\frac{81 x^{2}}{81 x^{2}} \cdot \frac{\frac{1}{9} - \frac{1}{x}}{\frac{1}{81} - \frac{1}{x^{2}}}$

Combine.

$\frac{81 x^{2} (\frac{1}{9} - \frac{1}{x})}{81 x^{2} (\frac{1}{81} - \frac{1}{x^{2}})}$

Apply the distributive property.

$\frac{81 x^{2} \frac{1}{9} + 81 x^{2} (- \frac{1}{x})}{81 x^{2} \frac{1}{81} + 81 x^{2} (- \frac{1}{x^{2}})}$

Simplify by cancelling.

Cancel the common factor of $9$ .

Factor $9$ out of $81 x^{2}$ .

$\frac{9 (9 x^{2}) \frac{1}{9} + 81 x^{2} (- \frac{1}{x})}{81 x^{2} \frac{1}{81} + 81 x^{2} (- \frac{1}{x^{2}})}$

Cancel the common factor.

$\frac{9 (9 x^{2}) \frac{1}{9} + 81 x^{2} (- \frac{1}{x})}{81 x^{2} \frac{1}{81} + 81 x^{2} (- \frac{1}{x^{2}})}$

Rewrite the expression.

$\frac{9 x^{2} + 81 x^{2} (- \frac{1}{x})}{81 x^{2} \frac{1}{81} + 81 x^{2} (- \frac{1}{x^{2}})}$

Cancel the common factor of $x$ .

Move the leading negative in $- \frac{1}{x}$ into the numerator.

$\frac{9 x^{2} + 81 x^{2} \frac{- 1}{x}}{81 x^{2} \frac{1}{81} + 81 x^{2} (- \frac{1}{x^{2}})}$

Factor $x$ out of $81 x^{2}$ .

$\frac{9 x^{2} + x (81 x) \frac{- 1}{x}}{81 x^{2} \frac{1}{81} + 81 x^{2} (- \frac{1}{x^{2}})}$

Cancel the common factor.

$\frac{9 x^{2} + x (81 x) \frac{- 1}{x}}{81 x^{2} \frac{1}{81} + 81 x^{2} (- \frac{1}{x^{2}})}$

Rewrite the expression.

$\frac{9 x^{2} + 81 x \cdot - 1}{81 x^{2} \frac{1}{81} + 81 x^{2} (- \frac{1}{x^{2}})}$

Multiply $- 1$ by $81$ .

$\frac{9 x^{2} - 81 x}{81 x^{2} \frac{1}{81} + 81 x^{2} (- \frac{1}{x^{2}})}$

Cancel the common factor of $81$ .

Factor $81$ out of $81 x^{2}$ .

$\frac{9 x^{2} - 81 x}{81 (x^{2}) \frac{1}{81} + 81 x^{2} (- \frac{1}{x^{2}})}$

Cancel the common factor.

$\frac{9 x^{2} - 81 x}{81 x^{2} \frac{1}{81} + 81 x^{2} (- \frac{1}{x^{2}})}$

Rewrite the expression.

$\frac{9 x^{2} - 81 x}{x^{2} + 81 x^{2} (- \frac{1}{x^{2}})}$

Cancel the common factor of $x^{2}$ .

Move the leading negative in $- \frac{1}{x^{2}}$ into the numerator.

$\frac{9 x^{2} - 81 x}{x^{2} + 81 x^{2} \frac{- 1}{x^{2}}}$

Factor $x^{2}$ out of $81 x^{2}$ .

$\frac{9 x^{2} - 81 x}{x^{2} + x^{2} \cdot 81 \frac{- 1}{x^{2}}}$

Cancel the common factor.

$\frac{9 x^{2} - 81 x}{x^{2} + x^{2} \cdot 81 \frac{- 1}{x^{2}}}$

Rewrite the expression.

$\frac{9 x^{2} - 81 x}{x^{2} + 81 \cdot - 1}$

Multiply $81$ by $- 1$ .

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

Factor $9 x$ out of $9 x^{2} - 81 x$ .

Factor $9 x$ out of $9 x^{2}$ .

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

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

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

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

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

Simplify the denominator.

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

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

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 $b = 9$ .

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

Cancel the common factor of $x - 9$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{9 x}{x + 9}$

Simplify (1/9-1/x)/(1/81-1/(x^2))

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