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Question Number 182120 by amin96 last updated on 04/Dec/22

Answered by SEKRET last updated on 04/Dec/22

$$\sqrt[4]{5+\frac{3}{\sqrt[6]{13+\frac{8}{....}}}}=\mathbf{a}$$$$5+\frac{3}{\sqrt[6]{13+\frac{8}{\mathbf{a}}}}={\mathbf{a}}^4$$$$(13+\frac{8}{\mathbf{a}})=\frac{3^6}{{({\mathbf{a}}^4-1)}^6}$$$$\mathbf{a}=\frac{\sqrt{5}+1}{2}$$

Answered by Frix last updated on 04/Dec/22

$$y=\sqrt[4]{5+\frac{3}{\sqrt[6]{13+\frac{8}{y}}}}$$$$\mathrm{Simply}\mathrm{because}\mathrm{this}\mathrm{is}\mathrm{generally}\mathrm{unsolvable}$$$$\mathrm{if}y\mathrm{is}\mathrm{no}‘‘{\mathrm{special}}^{″}\mathrm{value}\mathrm{I}\mathrm{tried}\mathrm{a}\mathrm{few}\mathrm{of}\mathrm{the}$$$$\mathrm{usual}‘‘{\mathrm{special}}^{″}\mathrm{values}\mathrm{and}\mathrm{found}$$$$y=\phi =\frac{1+\sqrt{5}}{2}$$

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