Irrational exponents

$we\quad have\quad to\quad check\quad the\quad greatest\quad the\quad graetest\\ will\quad remain\quad greatest\quad if\quad we\quad multiply\\there\quad powers\\ by\quad same\quad +ve\quad numbers\\ so:::\\ \quad \quad \quad \quad \quad { e }^{ \pi }\quad and\quad { \pi }^{ e }\\ { =e }^{ \pi /\pi e }\quad and\quad { \pi }^{ e/e\pi }\quad \\ ={ e }^{ 1/e }{ and\quad \pi }^{ 1/\pi }\\ let\quad us\quad considor\quad a\quad function\\ f\left( x \right) ={ x }^{ 1/x }\\ then\quad f^{ ' }\left( x \right) =\frac { { x }^{ 1/x } }{ { x }^{ 2 } } (1-lnx)\\ then\quad function\quad will\quad increase\quad till\quad x=e\quad then\quad decrease\\ \Longrightarrow \quad function\quad at\quad x=\quad e>\pi ;\\ \Longrightarrow { e }^{ 1/e }>{ \pi }^{ 1/\pi }\\ \Longrightarrow { e }^{ \pi }>{ \pi }^{ e }$ Image Credit :: Desmos.com