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Question Number 189559 by 073 last updated on 18/Mar/23

(−1)^(1/3) =? ((−1))^(1/3) =?

$$\:\left(−\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} =? \\ $$$$\sqrt[{\mathrm{3}}]{−\mathrm{1}}=? \\ $$

Answered by Frix last updated on 18/Mar/23

−1=e^(iπ) ⇒ (−1)^(1/n) =e^(i(π/n)) =cos (π/n) +i sin (π/n) (−1)^(1/2) =e^(i(π/2)) =cos (π/2) +i sin (π/2) =i (−1)^(1/3) =cos (π/3) +i sin (π/3) =(1/2)+((√3)/2)i

$$−\mathrm{1}=\mathrm{e}^{\mathrm{i}\pi} \\ $$$$\Rightarrow \\ $$$$\left(−\mathrm{1}\right)^{\frac{\mathrm{1}}{{n}}} =\mathrm{e}^{\mathrm{i}\frac{\pi}{{n}}} =\mathrm{cos}\:\frac{\pi}{{n}}\:+\mathrm{i}\:\mathrm{sin}\:\frac{\pi}{{n}} \\ $$$$\left(−\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{e}^{\mathrm{i}\frac{\pi}{\mathrm{2}}} =\mathrm{cos}\:\frac{\pi}{\mathrm{2}}\:+\mathrm{i}\:\mathrm{sin}\:\frac{\pi}{\mathrm{2}}\:=\mathrm{i} \\ $$$$\left(−\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{3}}} =\mathrm{cos}\:\frac{\pi}{\mathrm{3}}\:+\mathrm{i}\:\mathrm{sin}\:\frac{\pi}{\mathrm{3}}\:=\frac{\mathrm{1}}{\mathrm{2}}+\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\mathrm{i} \\ $$

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