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Question Number 201192 by Euclid last updated on 01/Dec/23

Answered by Calculusboy last updated on 01/Dec/23

$$\mathit{S}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}:\mathit{A}\mathit{p}\mathit{p}\mathit{l}\mathit{y}\mathit{I}\mathbf{n}\mathbf{to}\mathit{b}\mathit{o}\mathit{t}\mathit{h}\mathit{s}\mathit{i}\mathit{d}\mathit{e}\mathit{s}$$$$\mathit{I}\mathit{n}{\mathit{x}}^{{\mathit{x}}^4}=\mathit{I}\mathit{n}64$$$${\mathit{x}}^4\mathit{I}\mathit{n}\mathit{x}=\mathit{I}\mathit{n}64$$$$\mathit{I}\mathit{n}\mathit{x}\cdot {\mathit{e}}^{4\mathit{I}\mathit{n}\mathit{x}}=\mathit{I}\mathit{n}64[\mathit{N}\mathit{b}:{\mathit{x}}^4={\mathit{e}}^{4\mathit{I}\mathit{n}\mathit{x}}]$$$$\mathit{m}\mathit{u}\mathit{l}\mathit{t}\mathit{i}\mathit{p}\mathit{l}\mathit{y}\mathit{t}\mathit{h}\mathit{r}\mathit{o}\mathit{u}\mathit{g}\mathit{h}\mathit{b}\mathit{y}4$$$$4\mathit{I}\mathit{n}\mathit{x}\cdot {\mathit{e}}^{4\mathit{I}\mathit{n}\mathit{x}}=4\mathit{I}\mathit{n}64$$$$\mathit{a}\mathit{p}\mathit{p}\mathit{l}\mathit{y}\mathit{l}\mathit{a}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\mathit{t}\mathit{f}\mathit{u}\mathit{n}\mathit{c}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{t}\mathit{o}\mathit{b}\mathit{o}\mathit{t}\mathit{h}\mathit{s}\mathit{i}\mathit{d}\mathit{e}\mathit{s}$$$$\mathit{W}[4\mathit{I}\mathit{n}\mathit{x}\cdot {\mathit{e}}^{4\mathit{I}\mathit{n}\mathit{x}}]=\mathit{W}\left[4\mathit{I}\mathit{n}{2}^6\right]$$$$\mathit{W}[4\mathit{I}\mathit{n}\mathit{x}\cdot {\mathit{e}}^{4\mathit{I}\mathit{n}\mathit{x}}]=\mathit{W}[4\cdot 6\mathit{I}\mathit{n}2]\iff \mathit{W}\left[24\mathit{I}\mathit{n}2\right]$$$$\mathit{W}[4\mathit{I}\mathit{n}\mathit{x}\cdot {\mathit{e}}^{4\mathit{I}\mathit{n}\mathit{x}}]=\mathit{W}[8\cdot 3\mathit{I}\mathit{n}2]\iff \mathit{W}\left[8\mathit{I}\mathit{n}{2}^3\right]$$$$\mathit{W}[4\mathit{I}\mathit{n}\mathit{x}\cdot {\mathit{e}}^{4\mathit{I}\mathit{n}\mathit{x}}]=\mathit{W}\left[8\mathit{I}\mathit{n}8\right]$$$$\mathit{N}\mathit{b}:\mathit{W}\left({\mathit{x}\mathit{e}}^{\mathit{x}}\right)=\mathit{x}\mathit{a}\mathit{n}\mathit{d}\mathit{W}\left(\mathit{a}\mathit{I}\mathit{n}\mathit{a}\right)=\mathit{I}\mathit{n}\mathit{a}$$$$4\mathit{I}\mathit{n}\mathit{x}=\mathit{I}\mathit{n}8$$$${\mathit{I}\mathit{n}\mathit{x}}^4=\mathit{I}\mathit{n}8$$$${\mathit{x}}^4=8$$$$\mathit{x}=\sqrt[4]{8}$$$$$$

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