Question

If $m\ne n$ and $(m+n{)}^{-1}\times (m^{-1}+n^{-1})=m^x{n}^y$, then $x+y$ is equal to

• A. 2
• B. 0
• C. -2
• D. 1

Answer 1

The correct option is C

-2

Given

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

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

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

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

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

$\Rightarrow x=-1andy=-1$

$\therefore x+y=-2$