Algebra Examples

${(x^{- 2} - y^{- 2})}^{- 1}$

Step 1

Simplify each term.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

Rewrite $\frac{1^{2}}{x^{2}}$ as ${(\frac{1}{x})}^{2}$ .

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

Step 3.3

Rewrite $\frac{1}{y^{2}}$ as ${(\frac{1}{y})}^{2}$ .

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

Step 3.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{1}{x}$ and $b = \frac{1}{y}$ .

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

Step 3.5

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 3.6

To write $\frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3.7

Write each expression with a common denominator of $x y$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.7.1

Multiply $\frac{1}{x}$ by $\frac{y}{y}$ .

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

Step 3.7.2

Multiply $\frac{1}{y}$ by $\frac{x}{x}$ .

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

Step 3.7.3

Reorder the factors of $y x$ .

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

Step 3.8

Combine the numerators over the common denominator.

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

Step 3.9

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

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

Step 3.10

To write $- \frac{1}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3.11

Write each expression with a common denominator of $x y$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.11.1

Multiply $\frac{1}{x}$ by $\frac{y}{y}$ .

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

Step 3.11.2

Multiply $\frac{1}{y}$ by $\frac{x}{x}$ .

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

Step 3.11.3

Reorder the factors of $y x$ .

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

Step 3.12

Combine the numerators over the common denominator.

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

Step 4

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

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

Add $1$ and $1$ .

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

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

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