Question Number 194021 by mr W last updated on 26/Jun/23
$${find}\:\sqrt[{\mathrm{3}}]{\mathrm{1}+\sqrt[{\mathrm{3}}]{\mathrm{2}}+\sqrt[{\mathrm{3}}]{\mathrm{4}}}=? \\ $$
Commented by BaliramKumar last updated on 26/Jun/23
$$\left(\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{3}}} −\mathrm{1}\right)^{−\frac{\mathrm{1}}{\mathrm{3}}} \: \\ $$
Answered by Skabetix last updated on 26/Jun/23
$$=\left(\left(\mathrm{2}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} +\left(\mathrm{4}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} +\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} \approx\mathrm{1}.\mathrm{57} \\ $$
Answered by witcher3 last updated on 26/Jun/23
$$\mathrm{x}=\sqrt[{\mathrm{3}}]{\mathrm{2}} \\ $$$$\mathrm{1}+\mathrm{x}+\mathrm{x}^{\mathrm{2}} =\frac{\mathrm{x}^{\mathrm{3}} −\mathrm{1}}{\mathrm{x}−\mathrm{1}}=\frac{\mathrm{1}}{\mathrm{x}−\mathrm{1}} \\ $$$$=\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{3}}} −\mathrm{1}}} \\ $$$$ \\ $$