Question

If $m\ne n$ then $(m+n{)}^{-1}\times (m^{-1}+n^{-1})=$ ______

• A. $mn$
• B. 0
• C. $m^{-1}{n}^{-1}$
• D. $m^{-2}{n}^{-2}$

Answer 1

The correct option is C

$m^{-1}{n}^{-1}$

Given

$(m+n{)}^{-1}\times (m^{-1}+n^{-1})$,

$\Rightarrow \frac{1}{m+n}\times (\frac{1}{m}+\frac{1}{n})$ ($\because a^{-m}=\frac{1}{a^m}$)

$\Rightarrow \frac{1}{m+n}\times \left(\frac{m+n}{nm}\right)$

$\Rightarrow \frac{m+n}{(m+n)mn}$ = $\frac{1}{mn}=(mn{)}^{-1}$

$\Rightarrow (mn{)}^{-1}=m^{-1}{n}^{-1}$ ($\because (ab{)}^m=a^m\times b^m$)