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Question-182120




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

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