Irrational exponents

Algebra Level 2

e π e^{\pi} or π e {\pi}^e , which one is greater?

Both are equal e π e^\pi π e \pi^e Both are undefined

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1 solution

Kandarp Singh
Jul 1, 2015

w e h a v e t o c h e c k t h e g r e a t e s t t h e g r a e t e s t w i l l r e m a i n g r e a t e s t i f w e m u l t i p l y t h e r e p o w e r s b y s a m e + v e n u m b e r s s o : : : e π a n d π e = e π / π e a n d π e / e π = e 1 / e a n d π 1 / π l e t u s c o n s i d o r a f u n c t i o n f ( x ) = x 1 / x t h e n f ( x ) = x 1 / x x 2 ( 1 l n x ) t h e n f u n c t i o n w i l l i n c r e a s e t i l l x = e t h e n d e c r e a s e f u n c t i o n a t x = e > π ; e 1 / e > π 1 / π e π > π e 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

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