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Question Number 182120 by amin96 last updated on 04/Dec/22
Answered by SEKRET last updated on 04/Dec/22
$$\:\:\:\sqrt[{\mathrm{4}}]{\mathrm{5}+\frac{\mathrm{3}}{\:\sqrt[{\mathrm{6}}]{\mathrm{13}+\frac{\mathrm{8}}{....}}}\:}\:\:\:\:=\:\:\boldsymbol{\mathrm{a}} \\ $$$$\:\:\:\:\:\:\mathrm{5}\:+\frac{\mathrm{3}}{\:\sqrt[{\mathrm{6}}]{\mathrm{13}+\frac{\mathrm{8}}{\boldsymbol{\mathrm{a}}}}}\:\:=\:\boldsymbol{\mathrm{a}}^{\mathrm{4}} \\ $$$$\:\:\:\left(\mathrm{13}+\frac{\mathrm{8}}{\boldsymbol{\mathrm{a}}}\right)=\:\frac{\mathrm{3}^{\mathrm{6}} }{\left(\boldsymbol{\mathrm{a}}^{\mathrm{4}} −\mathrm{1}\right)^{\mathrm{6}} } \\ $$$$\:\:\:\boldsymbol{\mathrm{a}}=\:\frac{\sqrt{\mathrm{5}}\:+\mathrm{1}}{\mathrm{2}} \\ $$
Answered by Frix last updated on 04/Dec/22
$${y}=\sqrt[{\mathrm{4}}]{\mathrm{5}+\frac{\mathrm{3}}{\:\sqrt[{\mathrm{6}}]{\mathrm{13}+\frac{\mathrm{8}}{{y}}}}} \\ $$$$\mathrm{Simply}\:\mathrm{because}\:\mathrm{this}\:\mathrm{is}\:\mathrm{generally}\:\mathrm{unsolvable} \\ $$$$\mathrm{if}\:{y}\:\mathrm{is}\:\mathrm{no}\:``\mathrm{special}''\:\mathrm{value}\:\mathrm{I}\:\mathrm{tried}\:\mathrm{a}\:\mathrm{few}\:\:\mathrm{of}\:\mathrm{the} \\ $$$$\mathrm{usual}\:``\mathrm{special}''\:\mathrm{values}\:\mathrm{and}\:\mathrm{found} \\ $$$${y}=\varphi=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$