Simplify :

(a^-1 + b^-1) ÷ (a^-2 - b^-2)

Given,

(a^-1 + b^-1) ÷ (a^-2 - b^-2)

Simplifying the expression :

$\Rightarrow (a^{-1} + b^{-1}) ÷ (a^{-2} - b^{-2}) \\[1em] \Rightarrow \Big(\dfrac{1}{a} + \dfrac{1}{b}\Big) ÷ \Big(\dfrac{1}{a^2} - \dfrac{1}{b^2}\Big) \\[1em] \Rightarrow \Big(\dfrac{b + a}{ab}\Big) ÷ \Big(\dfrac{b^2 - a^2}{a^2b^2}\Big) \\[1em] \Rightarrow \Big(\dfrac{b + a}{ab}\Big) \times \dfrac{a^2b^2}{b^2 - a^2}\\[1em] \Rightarrow \dfrac{ab(b + a)}{(b + a)(b - a)} \\[1em] \Rightarrow \dfrac{ab}{(b - a)}.$

Hence, $(a^{-1} + b^{-1}) ÷ (a^{-2} - b^{-2}) = \dfrac{ab}{(b - a)}$.