Simplify (x^-1-y^-1)/(x^-2-y^-2)

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

Step 1

Simplify the numerator.

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

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

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

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

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

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

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

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

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

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

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

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

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

Reorder the factors of $y x$ .

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

Combine the numerators over the common denominator.

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

Step 2

Simplify the denominator.

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

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

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

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

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 $b = y^{- 1}$ .

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

Simplify.

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

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

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

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

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

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

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

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

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

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

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

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

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

Reorder the factors of $y x$ .

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

Combine the numerators over the common denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Reorder the factors of $y x$ .

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

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.

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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 $y - x$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

Cancel the common factor of $x y$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify (x^-1-y^-1)/(x^-2-y^-2)

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