Question Number 160871 by cortano last updated on 08/Dec/21
$$\:\:\sqrt[{\mathrm{6}}]{\left(\mathrm{2}\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{2}\sqrt{\mathrm{13}}+\mathrm{5}}{\:\sqrt{\mathrm{5}}+\mathrm{2}}}\:+\mathrm{2}\:\sqrt[{\mathrm{3}}]{\frac{\mathrm{2}\sqrt{\mathrm{13}}−\mathrm{5}}{\:\sqrt{\mathrm{5}}−\mathrm{2}}}+\mathrm{1}\right)^{\mathrm{2}} −\mathrm{1}}=? \\ $$
Answered by MJS_new last updated on 08/Dec/21
$$\left(\pm\mathrm{5}+\mathrm{2}\sqrt{\mathrm{13}}\right)^{\mathrm{1}/\mathrm{3}} =\pm\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{13}}}{\mathrm{2}} \\ $$$$\left(\pm\mathrm{2}+\sqrt{\mathrm{5}}\right)^{\mathrm{1}/\mathrm{3}} =\pm\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\Rightarrow\:\mathrm{answer}\:\mathrm{is}\:\mathrm{2} \\ $$