Question Number 86230 by M±th+et£s last updated on 27/Mar/20
$${is}\:\:\left(−\mathrm{1}\right)^{\frac{{m}}{{n}}} \:=\left(\sqrt[{{n}}]{\left(−\mathrm{1}\:\right)^{{m}} }\right)\:{or}\:=\left(\sqrt[{{n}}]{−\mathrm{1}}\right)^{{m}} \\ $$$${or}\:{both}\:{of}\:{them}\:{are}\:{fault}\:{and}\:{why}\:? \\ $$
Answered by MJS last updated on 27/Mar/20
$$\left(−\mathrm{1}\right)^{\frac{{m}}{{n}}} =\left(\mathrm{e}^{\mathrm{i}\pi} \right)^{\frac{{m}}{{n}}} =\mathrm{e}^{\mathrm{i}\pi\frac{{m}}{{n}}} =\mathrm{cos}\:\frac{\pi{m}}{{n}}\:+\mathrm{i}\:\mathrm{sin}\:\frac{\pi{m}}{{n}} \\ $$$$\sqrt[{{n}}]{\left(−\mathrm{1}\right)^{{m}} }=\sqrt[{{n}}]{\left(\mathrm{e}^{\mathrm{i}\pi} \right)^{{m}} }=\sqrt[{{n}}]{\mathrm{e}^{\mathrm{i}\pi{m}} }=\begin{cases}{\sqrt[{{n}}]{\mathrm{1}}=\mathrm{1};\:{m}=\mathrm{2}{k}}\\{\sqrt[{{n}}]{−\mathrm{1}}=\mathrm{e}^{\mathrm{i}\frac{\pi}{{n}}} =\mathrm{cos}\:\frac{\pi}{{n}}\:+\mathrm{i}\:\mathrm{sin}\:\frac{\pi}{{n}};\:{m}=\mathrm{2}{k}+\mathrm{1}}\end{cases};\:{k}\in\mathbb{Z} \\ $$$$\left(\sqrt[{{n}}]{−\mathrm{1}}\right)^{{m}} =\left(\sqrt[{{n}}]{\mathrm{e}^{\mathrm{i}\pi} }\right)^{{m}} =\left(\mathrm{e}^{\mathrm{i}\frac{\pi}{{n}}} \right)^{{m}} =\mathrm{e}^{\mathrm{i}\frac{\pi{m}}{{n}}} =\mathrm{cos}\:\frac{\pi{m}}{{n}}\:+\mathrm{i}\:\mathrm{sin}\:\frac{\pi{m}}{{n}} \\ $$
Commented by M±th+et£s last updated on 27/Mar/20
$${thank}\:{you}\:{sir}\: \\ $$