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

$$2^{\frac{1}{x}}=\sqrt{x}$$$$Findx$$

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

$$x=2$$

Answered by MrW3 last updated on 28/Aug/18

$$x>0$$$$2^{\frac{2}{x}}=x$$$$e^{\frac{2}{x}\ln 2}=x$$$$\frac{1}{x}{e}^{\frac{2}{x}\ln 2}=1$$$$\left(\frac{2}{x}\ln 2\right)e^{\left(\frac{2}{x}\ln 2\right)}=2\ln 2$$$$\Rightarrow \frac{2}{x}\ln 2=\mathbb{W}\left(2\ln 2\right)$$$$\Rightarrow x=\frac{2\ln 2}{\mathbb{W}\left(2\ln 2\right)}=\frac{2\ln 2}{0.6931}=2$$

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