In a lecture, our sir gave us a question comparing 2 constant values that were all of the form $α^β$ and $β^α$. Example ($π^e$, $e^π$) and ($2008^{2009}$, $2009^{2008}$). He used variable assumptions:
$α^{1/α}$ = $a^{1/ab}$ and $β^{1/β}$ = $b^{1/ab}$
After this, I formed the function f(x) = $x^{1/x}$ and checked it’s monotonicity but I can’t reach any result. Is that the correct method? And is there any better/other method to compare such values i.e. values of the form $α^β$ and $β^α$? If yes, please explain.