Question

If $x=2$ and $y=4$, find value of ${\left(\frac{x}{y}\right)}^{x-y}+{\left(\frac{y}{x}\right)}^{y-x}$.

Answer 1

Given expression is ${\left(\frac{x}{y}\right)}^{x-y}+{\left(\frac{y}{x}\right)}^{y-x}$
Substitute $x=2$ and $y=4$ in above expression,
$={\left(\frac{2}{4}\right)}^{-2}+{\left(\frac{4}{2}\right)}^2$
$={\left(\frac{1}{2}\right)}^{-2}+2^2$
$=(2{)}^2+4$ [Since, ${\left(\frac{1}{a}\right)}^{-m}=a^m$]
$=4+4$
$=8$
Hence, the value of the given expression is $8$.