Question Number 197299 by bbbbbbbb last updated on 12/Sep/23
$$\sqrt[{\mathrm{6}}]{−\mathrm{64}}−\sqrt[{\mathrm{10}}]{−\mathrm{10}}=? \\ $$$$\boldsymbol{{I}}\:\boldsymbol{\mathrm{need}}\:\boldsymbol{\mathrm{so}}\:\boldsymbol{\mathrm{much}}\:\boldsymbol{\mathrm{plz}} \\ $$
Answered by Frix last updated on 13/Sep/23
$$\forall{n}\in\mathbb{N}\backslash\left\{\mathrm{0}\right\}:\:−{n}={n}\mathrm{e}^{\mathrm{i}\pi} \:\Rightarrow \\ $$$$\sqrt[{{k}}]{−{n}}=\sqrt[{{k}}]{{n}}\mathrm{e}^{\mathrm{i}\frac{\pi}{{k}}} =\sqrt[{{k}}]{{n}}\left(\mathrm{cos}\:\frac{\pi}{{k}}\:+\mathrm{i}\:\mathrm{sin}\:\frac{\pi}{{k}}\right) \\ $$