Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $y$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x$ .

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Step 3.4.1

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

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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 $b = x$ .

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

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

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

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Step 5.2.1

Reorder terms.

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

$\frac{y - x}{x y}$