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Question Number 166241 by MathsFan last updated on 15/Feb/22

$${\mathit{x}}^5-1=0$$$$\mathit{p}\mathit{l}\mathit{e}\mathit{a}\mathit{s}\mathit{e}\mathit{h}\mathit{o}\mathit{w}\mathit{d}\mathit{o}\mathit{i}\mathit{f}\mathit{i}\mathit{n}\mathit{d}\mathit{f}\mathit{o}\mathit{r}\mathit{a}\mathit{l}\mathit{l}\mathit{t}\mathit{h}\mathit{e}$$$$\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathit{s}\mathit{o}\mathit{f}\mathit{x}?$$

Answered by batamatikayoga last updated on 16/Feb/22

$$x^5=1=e^{2\pi ik},k\in \{1,2,3,4,5\}$$$$x=e^{\frac{2}{5}\pi ik}$$

Commented by MathsFan last updated on 16/Feb/22

$$1+e^{\pi i}=0$$$$1=-e^{\pi i}????$$

Commented by alephzero last updated on 16/Feb/22

$$yes$$$$1=-e^{i\pi }$$$$\Rightarrow -1=-(-e^{i\pi })=e^{i\pi }$$

Answered by JDamian last updated on 16/Feb/22

$$x=e^{\frac{2\pi i}{5}k}\mathrm{\forall }k\in \{0,1,2,3,4\}$$

Commented by MathsFan last updated on 16/Feb/22

$$thankyousir$$

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