Simplify :

(a + b)^-1 × (a^-1 + b^-1)

Given,

(a + b)^-1 × (a^-1 + b^-1)

Simplifying the expression :

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

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