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Question Number 175834 by Linton last updated on 08/Sep/22
xe^x^(1/)  = e  solve for x
$${xe}^{\overset{\mathrm{1}/} {{x}}} =\:{e} \\ $$$${solve}\:{for}\:{x} \\ $$
Commented by mr W last updated on 08/Sep/22
put your question in order!  do you mean xe^(1/x) =e?  then it′s clear x=1.
$${put}\:{your}\:{question}\:{in}\:{order}! \\ $$$${do}\:{you}\:{mean}\:{xe}^{\frac{\mathrm{1}}{{x}}} ={e}? \\ $$$${then}\:{it}'{s}\:{clear}\:{x}=\mathrm{1}. \\ $$
Commented by Linton last updated on 08/Sep/22
please show working
$${please}\:{show}\:{working} \\ $$
Commented by mr W last updated on 08/Sep/22
if the question is xe^(1/x) =e, then it′s  obvious that x=1.  for xe^(1/x) =3, the solution is not obvious,  but it can be solved using lambert W  function, see below.
$${if}\:{the}\:{question}\:{is}\:{xe}^{\frac{\mathrm{1}}{{x}}} ={e},\:{then}\:{it}'{s} \\ $$$${obvious}\:{that}\:{x}=\mathrm{1}. \\ $$$${for}\:{xe}^{\frac{\mathrm{1}}{{x}}} =\mathrm{3},\:{the}\:{solution}\:{is}\:{not}\:{obvious}, \\ $$$${but}\:{it}\:{can}\:{be}\:{solved}\:{using}\:{lambert}\:{W} \\ $$$${function},\:{see}\:{below}. \\ $$
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)) :}
$${xe}^{\frac{\mathrm{1}}{{x}}} =\mathrm{3} \\ $$$$\frac{\mathrm{1}}{{x}}{e}^{−\frac{\mathrm{1}}{{x}}} =\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$\left(−\frac{\mathrm{1}}{{x}}\right){e}^{−\frac{\mathrm{1}}{{x}}} =−\frac{\mathrm{1}}{\mathrm{3}} \\ $$$$−\frac{\mathrm{1}}{{x}}={W}\left(−\frac{\mathrm{1}}{\mathrm{3}}\right) \\ $$$$\Rightarrow{x}=−\frac{\mathrm{1}}{{W}\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)} \\ $$$$=−\frac{\mathrm{1}}{−\mathrm{1}.\mathrm{512135}\:{or}\:−\mathrm{0}.\mathrm{619061}}=\begin{cases}{\mathrm{0}.\mathrm{661317}}\\{\mathrm{1}.\mathrm{615349}}\end{cases} \\ $$

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