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Question Number 143324 by Rankut last updated on 13/Jun/21

$${\left({\mathit{x}}^{2{\mathit{x}}^{-\frac{1}{5}}}\right)}^{-1}=\frac{1}{25}$$$$\mathbf{solve}\mathbf{for}\mathbf{x}$$

Answered by gsk2684 last updated on 13/Jun/21

$${\mathrm{x}}^{{2\mathrm{x}}^{-\frac{1}{5}}}=25\Rightarrow {2\mathrm{x}}^{-\frac{1}{5}}\ln \mathrm{x}=\ln 25$$$${2\mathrm{x}}^{-\frac{1}{5}}\ln \mathrm{x}=2\ln 5$$$$\mathrm{satisfied}\mathrm{by}{5}^5$$

Answered by mr W last updated on 13/Jun/21

$$x^{2x^{-\frac{1}{5}}}=25$$$${\left(x^{-\frac{1}{5}}\right)}^{x^{-\frac{1}{5}}}={25}^{-\frac{1}{10}}=5^{-\frac{1}{5}}$$$$letu=x^{-\frac{1}{5}}$$$$u^u=5^{-\frac{1}{5}}$$$$e^{\ln u}\ln u=-\frac{\ln 5}{5}$$$$\ln u=W(-\frac{\ln 5}{5})$$$$-\frac{1}{5}\ln x=W(-\frac{\ln 5}{5})$$$$\Rightarrow x=e^{-5W(-\frac{\ln 5}{5})}$$$$\approx \{\begin{array}{l}{e}^{-5\times (-1.609437912)}=3125=5^5\\ e^{-5\times (-0.568064483)}=17.124878\end{array}$$

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