Question Number 182188 by mathlove last updated on 05/Dec/22
$$\left(\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}+\mathrm{2}}+\sqrt[{\mathrm{3}}]{\sqrt{\mathrm{5}}−\mathrm{2}}\right)^{\mathrm{2014}} =? \\ $$
Answered by Frix last updated on 05/Dec/22
$$\sqrt[{\mathrm{3}}]{\mathrm{2}+\sqrt{\mathrm{5}}}=\varphi\wedge\sqrt[{\mathrm{3}}]{−\mathrm{2}+\sqrt{\mathrm{5}}}=\frac{\mathrm{1}}{\varphi} \\ $$$$\varphi+\frac{\mathrm{1}}{\varphi}=\sqrt{\mathrm{5}} \\ $$$$\left(\sqrt{\mathrm{5}}\right)^{\mathrm{2014}} =\mathrm{5}^{\mathrm{1007}} \\ $$
Answered by mnjuly1970 last updated on 05/Dec/22
$$\:\:\:{x}^{\mathrm{3}} =\:\mathrm{2}\sqrt{\mathrm{5}}\:+\:\mathrm{3}\:{x}\:\Rightarrow\:{x}\:=\sqrt{\mathrm{5}} \\ $$$$\:\:\:\:{answer}\:=\:\mathrm{5}^{\:\mathrm{1007}} \\ $$