# For all values for which the fraction is defined, how do you simplify (a^-1-b^-1)/(b^-2-a^-2)?

Jan 7, 2017

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

#### Explanation:

Multiply both numerator and denominator by ${a}^{2} {b}^{2}$ and simplify as follows:

$\frac{{a}^{- 1} - {b}^{- 1}}{{b}^{- 2} - {a}^{- 2}} = \frac{{a}^{2} {b}^{2} \left({a}^{- 1} - {b}^{- 1}\right)}{{a}^{2} {b}^{2} \left({b}^{- 2} - {a}^{- 2}\right)}$

$\textcolor{w h i t e}{\frac{{a}^{- 1} - {b}^{- 1}}{{b}^{- 2} - {a}^{- 2}}} = \frac{{a}^{2} {b}^{2} {a}^{- 1} - {a}^{2} {b}^{2} {b}^{- 1}}{{a}^{2} {b}^{2} {b}^{- 2} - {a}^{2} {b}^{2} {a}^{- 2}}$

$\textcolor{w h i t e}{\frac{{a}^{- 1} - {b}^{- 1}}{{b}^{- 2} - {a}^{- 2}}} = \frac{a {b}^{2} - {a}^{2} b}{{a}^{2} - {b}^{2}}$

$\textcolor{w h i t e}{\frac{{a}^{- 1} - {b}^{- 1}}{{b}^{- 2} - {a}^{- 2}}} = \frac{a b \left(b - a\right)}{\left(a - b\right) \left(a + b\right)}$

$\textcolor{w h i t e}{\frac{{a}^{- 1} - {b}^{- 1}}{{b}^{- 2} - {a}^{- 2}}} = - \frac{a b \textcolor{red}{\cancel{\textcolor{b l a c k}{\left(a - b\right)}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{\left(a - b\right)}}} \left(a + b\right)}$

$\textcolor{w h i t e}{\frac{{a}^{- 1} - {b}^{- 1}}{{b}^{- 2} - {a}^{- 2}}} = - \frac{a b}{a + b}$