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Question Number 42550 by Cheyboy last updated on 27/Aug/18

2^(1/x) =(√x)  Find x

$$\mathrm{2}^{\frac{\mathrm{1}}{{x}}} =\sqrt{{x}} \\ $$$${Find}\:{x} \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 28/Aug/18

Commented by tanmay.chaudhury50@gmail.com last updated on 28/Aug/18

Commented by tanmay.chaudhury50@gmail.com last updated on 28/Aug/18

x=2

$${x}=\mathrm{2} \\ $$

Answered by MrW3 last updated on 28/Aug/18

x>0  2^(2/x) =x  e^((2/x) ln 2) =x  (1/x) e^((2/x) ln 2) =1  ((2/x) ln 2) e^(((2/x) ln 2)) =2 ln 2  ⇒(2/x) ln 2=W(2 ln 2)  ⇒x=((2 ln 2)/(W(2 ln 2)))=((2 ln 2)/(0.6931))=2

$${x}>\mathrm{0} \\ $$$$\mathrm{2}^{\frac{\mathrm{2}}{{x}}} ={x} \\ $$$${e}^{\frac{\mathrm{2}}{{x}}\:\mathrm{ln}\:\mathrm{2}} ={x} \\ $$$$\frac{\mathrm{1}}{{x}}\:{e}^{\frac{\mathrm{2}}{{x}}\:\mathrm{ln}\:\mathrm{2}} =\mathrm{1} \\ $$$$\left(\frac{\mathrm{2}}{{x}}\:\mathrm{ln}\:\mathrm{2}\right)\:{e}^{\left(\frac{\mathrm{2}}{{x}}\:\mathrm{ln}\:\mathrm{2}\right)} =\mathrm{2}\:\mathrm{ln}\:\mathrm{2} \\ $$$$\Rightarrow\frac{\mathrm{2}}{{x}}\:\mathrm{ln}\:\mathrm{2}=\mathbb{W}\left(\mathrm{2}\:\mathrm{ln}\:\mathrm{2}\right) \\ $$$$\Rightarrow{x}=\frac{\mathrm{2}\:\mathrm{ln}\:\mathrm{2}}{\mathbb{W}\left(\mathrm{2}\:\mathrm{ln}\:\mathrm{2}\right)}=\frac{\mathrm{2}\:\mathrm{ln}\:\mathrm{2}}{\mathrm{0}.\mathrm{6931}}=\mathrm{2} \\ $$

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