Precalculus Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $9 x^{3}$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x^{3}$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $9$ .

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Move $- 1$ to the left of $x^{3}$ .

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

Step 4.2

Rewrite $- 1 x^{3}$ as $- x^{3}$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $9 x$ out of $9 x + 9 x^{3} \cdot - 3$ .

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Step 5.1.1

Factor $9 x$ out of $9 x$ .

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

Step 5.1.2

Factor $9 x$ out of $9 x^{3} \cdot - 3$ .

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

Step 5.1.3

Factor $9 x$ out of $9 x (1) + 9 x (x^{2} \cdot - 3)$ .

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

Step 5.2

Move $- 3$ to the left of $x^{2}$ .

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