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Question Number 128015 by Algoritm last updated on 03/Jan/21

Answered by MJS_new last updated on 03/Jan/21

$$x=\sqrt[8]{2}$$

Answered by MJS_new last updated on 03/Jan/21

$$x^{x^{20}}=2^{\frac{1}{\sqrt{2}}}$$$$\mathrm{let}x=2^r[\mathrm{obviously}\mathrm{only}{2}^k-\mathrm{factors}\mathrm{on}\mathrm{the}\mathrm{rhs}]$$$$2^{2^{20r}r}=2^{\frac{1}{\sqrt{2}}}$$$$2^{20r}r=\frac{1}{\sqrt{2}}=2^{-\frac{1}{2}}$$$$\mathrm{again}\mathrm{only}{2}^k-\mathrm{factors}\Rightarrow r=2^{-n}$$$$\mathrm{now}\mathrm{try}n=1,2,3,...$$$$\Rightarrow n=3\Rightarrow r=\frac{1}{8}\Rightarrow x=\sqrt[8]{2}$$

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