For $z\ne0$, the principal value of <complex exponentiation> is $z^w=\exp(w\operatorname{Log}z)$, where the <principal complex logarithm> is $\operatorname{Log}z=\log|z|+i\operatorname{Arg}z$ with $-\pi<\operatorname{Arg}z\leq\pi$. <De Moivre's theorem> states that $(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta)$ for every <integer> $n$.

Since $\sqrt3+i=2e^{i\pi/6}$, the six roots are
$$
\boxed{z=2^{1/6}\exp\left(i\left(\frac{\pi}{36}+\frac{k\pi}{3}\right)\right),\qquad k=0,1,\ldots,5.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 1 / 1C / i / Solution

{c}

For every <integer> $n$ and real $\theta$,
$$
(\cos\theta+i\sin\theta)^n=\cos(n\theta)+i\sin(n\theta).
$$
It converts powers of a <complex number> in polar form into multiplication of its <complex argument>.

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 Incoming links: De Moivre's theorem