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Question Number 7178 by aftab ahmad last updated on 14/Aug/16

Commented by aftab ahmad last updated on 14/Aug/16

$$\mathit{V}\mathit{e}\mathit{r}\mathit{y}\mathit{n}\mathit{i}\mathit{c}\mathit{e}\mathit{a}\mathit{p}\mathit{p}\mathit{r}\mathit{o}\mathit{a}\mathit{c}\mathit{h}\mathit{s}\mathit{i}\mathit{r}..\mathit{T}h\mathit{a}\mathit{n}\mathit{k}\mathit{s}\mathit{a}\mathit{l}\mathit{o}\mathit{t}.$$

Commented by Yozzia last updated on 14/Aug/16

$$x^{x^{1/3}}=\frac{9^{1/3}}{3}=\frac{1}{3^{1/3}}$$$$Letu=x^{1/3}\Rightarrow x=u^3$$$$\therefore u^{3u}=3^{-1/3}$$$$\Rightarrow u^u=3^{-1/9}$$$$ulnu=ln{3}^{-1/9}$$$$e^{lnu}lnu=\frac{-1}{9}ln3$$$$\Rightarrow W\left(e^{lnu}lnu\right)=W\left(\frac{-1}{9}ln3\right)$$$$\Rightarrow lnu=W\left(\frac{-ln3}{9}\right)$$$$\Rightarrow u=e^{W\left(\frac{-ln3}{9}\right)}$$$$\therefore x^{1/3}=e^{W\left(\frac{-ln3}{9}\right)}$$$$x=e^{3W\left(\frac{-ln3}{9}\right)}\approx 0.6561$$$$W=product-logfunction$$$$$$

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