Question Number 204691 by Davidtim last updated on 25/Feb/24
$$\sqrt{{a}\boldsymbol{\div}\sqrt{{a}\boldsymbol{\div}\sqrt{{a}\boldsymbol{\div}\centerdot\centerdot\centerdot\boldsymbol{\div}\sqrt{{a}}}}}=? \\ $$
Answered by Faetmaaa last updated on 27/Feb/24
$$\forall{a}\in\boldsymbol{\mathrm{R}}\backslash\left\{\mathrm{0}\right\} \\ $$$${x}\::=\:\sqrt{\frac{{a}}{\:\sqrt{\frac{{a}}{\:\sqrt{\frac{{a}}{\ldots}}}}}} \\ $$$$\Rightarrow{x}=\sqrt{\frac{{a}}{{x}}} \\ $$$$\Rightarrow\:{x}^{\mathrm{2}} =\frac{{a}}{{x}} \\ $$$$\Rightarrow\:{x}^{\mathrm{3}} ={a} \\ $$$$\Rightarrow\:{x}=\sqrt[{\mathrm{3}}]{{a}} \\ $$