Algebra Examples

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

Step 1

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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Step 2.7.1

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

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

Step 2.7.2

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

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

Step 2.7.3

Reorder the factors of $y x$ .

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

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

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Step 2.11.1

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

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

Step 2.11.2

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

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

Step 2.11.3

Reorder the factors of $y x$ .

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

Step 2.12

Combine the numerators over the common denominator.

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

Step 2.13

Combine exponents.

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Step 2.13.1

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

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

Step 2.13.2

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

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

Step 2.13.3

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

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

Step 2.13.4

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

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

Step 2.13.5

Add $1$ and $1$ .

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

Step 2.13.6

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

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

Step 2.13.7

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

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

Step 2.13.8

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

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

Step 2.13.9

Add $1$ and $1$ .

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

Step 2.13.10

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

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

Step 2.14

Reduce the expression $\frac{(y + x) (y - x) x^{2}}{x^{2} y^{2}}$ by cancelling the common factors.

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Step 2.14.1

Cancel the common factor.

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

Step 2.14.2

Rewrite the expression.

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

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

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