<s>[INST] How to solve $x^3=-1$? How to solve $x^3=-1$? I got following: 
$x^3=-1$ 
$x=(-1)^{\frac{1}{3}}$
$x=\frac{(-1)^{\frac{1}{2}}}{(-1)^{\frac{1}{6}}}=\frac{i}{(-1)^{\frac{1}{6}}}$...
 [/INST] The real solution:
$$x^3+1=0<=>$$
$$x^3=-1<=>$$
Take cube roots of both sides:
$$x=-1$$
Complex solution:
$$x^3+1=0<=>$$
$$x^3=-1<=>$$
$$x^3=|-1|e^{arg(-1)i}<=>$$
$$x^3=e^{(\pi +2\pi k)i}<=>$$
(with k is the element of Z)
$$x=\left(e^{(\pi +2\pi k)i}\right)^{\frac{1}{3}}<=>$$
$$x=e^{\left(\frac{1}{3}\pi +\frac{2}{3}\pi k\right)i}$$
(with k goes from 0 to 3 -> k=0-2)
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