What is (a^{-1}+b^{-1})/(a^{-2)-b^{-2}} ?

The solution to (a^{-1}+b^{-1})/(a^{-2)-b^{-2}} is (ab)/(b-a)

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simplify (a^{-1}+b^{-1})/(a^{-2)-b^{-2}}

simplify

\frac{a ^{- 1} + b ^{- 1}}{a ^{- 2} - b ^{- 2}}

\frac{ab}{b - a}

Solution steps

\frac{a ^{- 1} + b ^{- 1}}{a ^{- 2} - b ^{- 2}}

Simplify

a^{- 2} - b^{- 2} : \frac{1}{a ^{2}} - \frac{1}{b ^{2}}

= \frac{a ^{- 1} + b ^{- 1}}{\frac{1}{a ^{2}} - \frac{1}{b ^{2}}}

Join

\frac{1}{a ^{2}} - \frac{1}{b ^{2}} : \frac{b ^{2} - a ^{2}}{a ^{2} b ^{2}}

= \frac{a ^{- 1} + b ^{- 1}}{\frac{b ^{2} - a ^{2}}{a ^{2} b ^{2}}}

Simplify

a^{- 1} + b^{- 1} : \frac{1}{a} + \frac{1}{b}

= \frac{\frac{1}{a} + \frac{1}{b}}{\frac{b ^{2} - a ^{2}}{a ^{2} b ^{2}}}

Join

\frac{1}{a} + \frac{1}{b} : \frac{b + a}{ab}

= \frac{\frac{b + a}{ab}}{\frac{b ^{2} - a ^{2}}{a ^{2} b ^{2}}}

Divide fractions:

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

= \frac{( b + a ) a ^{2} b ^{2}}{ab ( b ^{2} - a ^{2} )}

Cancel

\frac{( b + a ) a ^{2} b ^{2}}{ab ( b ^{2} - a ^{2} )} : \frac{ab ( a + b )}{b ^{2} - a ^{2}}

= \frac{ab ( a + b )}{b ^{2} - a ^{2}}

Apply Difference of Two Squares Formula:

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

b^{2} - a^{2} = (b + a) (b - a)

= \frac{ab ( a + b )}{( b + a ) ( b - a )}

Cancel the common factor:

a + b

= \frac{ab}{b - a}

3 x + 2 + x = x + 5

ln (x) - 6 ln (y) + ln (z)

8 = \frac{z}{6}

5 x^{2} + 22 x - 120 = 12 x

E, y^{2} - 2 y E - x^{2} + 2 x E \leq 0

What is (a^{-1}+b^{-1})/(a^{-2)-b^{-2}} ?
The solution to (a^{-1}+b^{-1})/(a^{-2)-b^{-2}} is (ab)/(b-a)