Precalculus Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $3 y^{2} x^{2}$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $y$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Add $1$ and $2$ .

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

Step 3.5

Cancel the common factor of $x$ .

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

Cancel the common factor.

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

Step 3.5.4

Rewrite the expression.

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

Step 3.6

Multiply $- 1$ by $3$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Add $2$ and $1$ .

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

Step 3.10

Cancel the common factor of $3 x^{2}$ .

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Step 3.10.1

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

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

Step 3.10.2

Cancel the common factor.

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

Step 3.10.3

Rewrite the expression.

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

Step 3.11

Cancel the common factor of $3 y^{2}$ .

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Step 3.11.1

Move the leading negative in $- \frac{1}{3 y^{2}}$ into the numerator.

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

Step 3.11.2

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

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

Step 3.11.3

Cancel the common factor.

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

Step 3.11.4

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

Factor $3 y x$ out of $3 y x (x^{2}) + 3 y x (- y^{2})$ .

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

Step 4.2

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 $b = y$ .

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

$\frac{3 y x (x + y) (x - y)}{(y + x) (y - x)}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $x + y$ and $y + x$ .

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Step 6.1.1

Reorder terms.

$\frac{3 y x (x + y) (x - y)}{(x + y) (y - x)}$

Step 6.1.2

Cancel the common factor.

$\frac{3 y x (x + y) (x - y)}{(x + y) (y - x)}$

Step 6.1.3

Rewrite the expression.

$\frac{3 y x (x - y)}{y - x}$

Step 6.2

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

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Step 6.2.1

Factor $- 1$ out of $x$ .

$\frac{3 y x (- 1 (- x) - y)}{y - x}$

Step 6.2.2

Factor $- 1$ out of $- y$ .

$\frac{3 y x (- 1 (- x) - (y))}{y - x}$

Step 6.2.3

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

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

Step 6.2.4

Reorder terms.

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

Step 6.2.5

Cancel the common factor.

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

Step 6.2.6

Divide $3 y x \cdot (- 1)$ by $1$ .

$3 y x \cdot (- 1)$

Step 6.3

Multiply $- 1$ by $3$ .

$- 3 y x$