An algebra problem by A Former Brilliant Member

(x/y)^(x/y)

=((x^x)/(y^x))^(1/y)

=((x^x)/(x^y))^(1/y)

.[beacuse, x^y=y^x]

=x^((x-y)*(1/y))

=x^((x/y)-1)

Given that, $x^y=y^x$ .. Now we have to get back to the term (\frac{x}{y})^\frac{x}{y} \Rightarrow (\frac{x^x}{y^x})^\frac{1}{y} \Rightarrow (\frac{x^x}{x^y})^\frac{1}{y} ; [plugging(\color\pink{y^x=x^y}] \Rightarrow (x^{x-y})^\frac{1}{y} \Rightarrow x^{\frac{x-y}{y}} \Rightarrow\boxed{\color\red{x^{\frac{x}{y} -1}}}