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Question Number 204691 by Davidtim last updated on 25/Feb/24

$$\sqrt{a\mathbf{÷}\sqrt{a\mathbf{÷}\sqrt{a\mathbf{÷}\cdot \cdot \cdot \mathbf{÷}\sqrt{a}}}}=?$$

Answered by Faetmaaa last updated on 27/Feb/24

$$\mathrm{\forall }a\in \mathbf{R}\mathrm{\setminus }\left\{0\right\}$$$$x:=\sqrt{\frac{a}{\sqrt{\frac{a}{\sqrt{\frac{a}{\dots }}}}}}$$$$\Rightarrow x=\sqrt{\frac{a}{x}}$$$$\Rightarrow x^2=\frac{a}{x}$$$$\Rightarrow x^3=a$$$$\Rightarrow x=\sqrt[3]{a}$$

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