Simplify (x/9-1/3)/(1/(x^2)-1/9)

$\frac{\frac{x}{9} - \frac{1}{3}}{\frac{1}{x^{2}} - \frac{1}{9}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $9 x^{2}$ .

Multiply $\frac{\frac{x}{9} - \frac{1}{3}}{\frac{1}{x^{2}} - \frac{1}{9}}$ by $\frac{9 x^{2}}{9 x^{2}}$ .

$\frac{9 x^{2}}{9 x^{2}} \cdot \frac{\frac{x}{9} - \frac{1}{3}}{\frac{1}{x^{2}} - \frac{1}{9}}$

Combine.

$\frac{9 x^{2} (\frac{x}{9} - \frac{1}{3})}{9 x^{2} (\frac{1}{x^{2}} - \frac{1}{9})}$

Step 2

Apply the distributive property.

$\frac{9 x^{2} \frac{x}{9} + 9 x^{2} (- \frac{1}{3})}{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

Step 3

Simplify by cancelling.

Cancel the common factor of $9$ .

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

$\frac{9 (x^{2}) \frac{x}{9} + 9 x^{2} (- \frac{1}{3})}{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

Cancel the common factor.

$\frac{9 x^{2} \frac{x}{9} + 9 x^{2} (- \frac{1}{3})}{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

Rewrite the expression.

$\frac{x^{2} x + 9 x^{2} (- \frac{1}{3})}{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

Multiply $x^{2}$ by $x$ by adding the exponents.

Multiply $x^{2}$ by $x$ .

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

$\frac{x^{2} x^{1} + 9 x^{2} (- \frac{1}{3})}{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

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

$\frac{x^{2 + 1} + 9 x^{2} (- \frac{1}{3})}{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

Add $2$ and $1$ .

$\frac{x^{3} + 9 x^{2} (- \frac{1}{3})}{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$\frac{x^{3} + 9 x^{2} \frac{- 1}{3}}{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

Factor $3$ out of $9 x^{2}$ .

$\frac{x^{3} + 3 (3 x^{2}) \frac{- 1}{3}}{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

Cancel the common factor.

$\frac{x^{3} + 3 (3 x^{2}) \frac{- 1}{3}}{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

Rewrite the expression.

$\frac{x^{3} + 3 x^{2} \cdot - 1}{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

Multiply $- 1$ by $3$ .

$\frac{x^{3} - 3 x^{2}}{9 x^{2} \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

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

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

$\frac{x^{3} - 3 x^{2}}{x^{2} \cdot 9 \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

Cancel the common factor.

$\frac{x^{3} - 3 x^{2}}{x^{2} \cdot 9 \frac{1}{x^{2}} + 9 x^{2} (- \frac{1}{9})}$

Rewrite the expression.

$\frac{x^{3} - 3 x^{2}}{9 + 9 x^{2} (- \frac{1}{9})}$

Cancel the common factor of $9$ .

Move the leading negative in $- \frac{1}{9}$ into the numerator.

$\frac{x^{3} - 3 x^{2}}{9 + 9 x^{2} \frac{- 1}{9}}$

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

$\frac{x^{3} - 3 x^{2}}{9 + 9 (x^{2}) \frac{- 1}{9}}$

Cancel the common factor.

$\frac{x^{3} - 3 x^{2}}{9 + 9 x^{2} \frac{- 1}{9}}$

Rewrite the expression.

$\frac{x^{3} - 3 x^{2}}{9 + x^{2} \cdot - 1}$

Step 4

Factor $x^{2}$ out of $x^{3} - 3 x^{2}$ .

Factor $x^{2}$ out of $x^{3}$ .

$\frac{x^{2} x - 3 x^{2}}{9 + x^{2} \cdot - 1}$

Factor $x^{2}$ out of $- 3 x^{2}$ .

$\frac{x^{2} x + x^{2} \cdot - 3}{9 + x^{2} \cdot - 1}$

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 3$ .

$\frac{x^{2} (x - 3)}{9 + x^{2} \cdot - 1}$

Step 5

Simplify the denominator.

Move $- 1$ to the left of $x^{2}$ .

$\frac{x^{2} (x - 3)}{9 - 1 \cdot x^{2}}$

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

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

Rewrite $9 - x^{2}$ in a factored form.

Rewrite $9$ as $3^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3$ and $b = x$ .

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

Step 6

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x - 3$ and $3 - x$ .

Factor $- 1$ out of $x$ .

$\frac{x^{2} (- 1 (- x) - 3)}{(3 + x) (3 - x)}$

Rewrite $- 3$ as $- 1 (3)$ .

$\frac{x^{2} (- 1 (- x) - 1 (3))}{(3 + x) (3 - x)}$

Factor $- 1$ out of $- 1 (- x) - 1 (3)$ .

$\frac{x^{2} (- 1 (- x + 3))}{(3 + x) (3 - x)}$

Reorder terms.

$\frac{x^{2} (- 1 (- x + 3))}{(3 + x) (- x + 3)}$

Cancel the common factor.

$\frac{x^{2} (- 1 (- x + 3))}{(3 + x) (- x + 3)}$

Rewrite the expression.

$\frac{x^{2} \cdot (- 1)}{3 + x}$

Simplify the expression.

Move $- 1$ to the left of $x^{2}$ .

$\frac{- 1 \cdot x^{2}}{3 + x}$

Move the negative in front of the fraction.

$- \frac{x^{2}}{3 + x}$

Simplify (x/9-1/3)/(1/(x^2)-1/9)

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