What is (x^{-1}-y^{-1})/(x^{-2)-y^{-2}} ?

The solution to (x^{-1}-y^{-1})/(x^{-2)-y^{-2}} is (xy)/(y+x)

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simplify (x^{-1}-y^{-1})/(x^{-2)-y^{-2}}

simplify

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

\frac{x y}{y + x}

Solution steps

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

Simplify

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

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

Join

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

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

Simplify

x^{- 1} - y^{- 1} : \frac{1}{x} - \frac{1}{y}

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

Join

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

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

Divide fractions:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

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

Cancel

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

y^{2} - x^{2} = (y + x) (y - x)

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

Cancel the common factor:

y - x

= \frac{x y}{y + x}

3 (x^{2} + 12) = 84

2 x^{2} = 17 x + 9

(e^{\frac{2 x}{3}})^{2}

\frac{x - 5}{x ^{2} + 3 x} > 0

(t + 15)^{2} + 5 = 0

What is (x^{-1}-y^{-1})/(x^{-2)-y^{-2}} ?
The solution to (x^{-1}-y^{-1})/(x^{-2)-y^{-2}} is (xy)/(y+x)