Pre-Algebra Examples

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

Step 1

Rewrite the division as a fraction.

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

Step 2

Simplify the numerator.

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Step 2.1

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

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

Step 2.2

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

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

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 $b = y^{- 1}$ .

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

Step 2.4

Simplify.

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Step 2.4.1

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

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

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

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

Step 2.5

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

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

Step 2.6

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

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

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 $\frac{y}{y}$ .

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

Step 2.7.2

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

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

Step 2.7.3

Reorder the factors of $y x$ .

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

Step 2.8

Combine the numerators over the common denominator.

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

Step 2.9

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

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

Step 2.10

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

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

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 $\frac{y}{y}$ .

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

Step 2.11.2

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 2.11.3

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 2.12

Combine the numerators over the common denominator.

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

Step 3

Simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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Step 3.5.1

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

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

Step 3.5.2

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

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

Step 3.5.3

Reorder the factors of $y x$ .

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

Step 3.6

Combine the numerators over the common denominator.

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

Step 4

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

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 5.5

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

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

Add $1$ and $1$ .

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

Cancel the common factor of $y + x$ .

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

Cancel the common factor of $x y$ .

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Step 8.1

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

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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

Step 9

Split the fraction $\frac{y - x}{x y}$ into two fractions.

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

Step 10

Cancel the common factor of $y$ .

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

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

Step 11

Cancel the common factor of $x$ .

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Step 11.1

Cancel the common factor.

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

Step 11.2

Rewrite the expression.

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

Step 12

Move the negative in front of the fraction.

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