Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

Reorder the factors of $y x$ .

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

Step 1.6

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

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 $b = y^{- 1}$ .

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

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

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

Step 2.4.5

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

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Step 2.4.5.1

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

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

Step 2.4.5.2

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

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

Step 2.4.5.3

Reorder the factors of $y x$ .

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

Step 2.4.6

Combine the numerators over the common denominator.

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

Step 2.4.7

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

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

Step 2.4.8

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

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

Step 2.4.9

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

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

Step 2.4.10

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

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

Step 2.4.11

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

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Step 2.4.11.1

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

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

Step 2.4.11.2

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

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

Step 2.4.11.3

Reorder the factors of $y x$ .

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

Step 2.4.12

Combine the numerators over the common denominator.

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

Step 3

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

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

Add $1$ and $1$ .

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

Step 5

Multiply the numerator by the reciprocal of the denominator.

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

Step 6

Cancel the common factor of $y + x$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

Cancel the common factor of $x y$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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