Prove that$ \left(a^{-1}+b^{-1}\right)^{-1}=\frac{a b}{a+b} $

Given:

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

To do:

We have to prove that \( \left(a^{-1}+b^{-1}\right)^{-1}=\frac{a b}{a+b} \).

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

$a^{m} \times a^{n}=a^{m+n}$

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$  

LHS $=(a^{-1}+b^{-1})^{-1}$

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

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

$=\frac{a b}{a+b}$

$=$ RHS

Hence proved.