xe^x^(1/) = e solve for x


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Question Number 175834 by Linton last updated on 08/Sep/22

${x e}^{\overset{1 /}{x}} = e$ $s o l v e f o r x$
Commented by mr W last updated on 08/Sep/22

$p u t y o u r q u e s t i o n i n o r d e r!$ $d o y o u m e a n {x e}^{\frac{1}{x}} = e ?$ $t h e n {i t}^{'} s c l e a r x = 1 .$
Commented by Linton last updated on 08/Sep/22

$p l e a s e s h o w w o r k i n g$
Commented by mr W last updated on 08/Sep/22

$i f t h e q u e s t i o n i s {x e}^{\frac{1}{x}} = e, t h e n {i t}^{'} s$ $o b v i o u s t h a t x = 1 .$ $f o r {x e}^{\frac{1}{x}} = 3, t h e s o l u t i o n i s n o t o b v i o u s,$ $b u t i t c a n b e s o l v e d u s i n g l a m b e r t W$ $f u n c t i o n, s e e b e l o w .$
Commented by mr W last updated on 08/Sep/22
$xe^(1/x) =3 (1/x)e^(−(1/x)) =(1/3) (−(1/x))e^(−(1/x)) =−(1/3) −(1/x)=W(−(1/3)) ⇒x=−(1/(W(−(1/3)))) =−(1/(−1.512135 or −0.619061))= { ((0.661317)),((1.615349)) :}$
${x e}^{\frac{1}{x}} = 3$ $\frac{1}{x} e^{- \frac{1}{x}} = \frac{1}{3}$ $(- \frac{1}{x}) e^{- \frac{1}{x}} = - \frac{1}{3}$ $- \frac{1}{x} = W (- \frac{1}{3})$ $\Rightarrow x = - \frac{1}{W (- \frac{1}{3})}$ $= - \frac{1}{- 1.512135 o r - 0.619061} = {\begin{cases} 0.661317 \\ 1.615349 \end{cases}$
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