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

$${(-1)}^{\frac{1}{3}}=?$$$$\sqrt[3]{-1}=?$$

Answered by Frix last updated on 18/Mar/23

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

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