Past exam of the mathematics course of the University of Cambridge / 2018 / ia / Paper 1 / 1C (Vectors and Matrices)

For $z\ne 0$, the principal value of complex exponentiation is $z^w=\exp (w\mathrm{Log}z)$, where the principal complex logarithm is $\mathrm{Log}z=\log |z|+i\mathrm{Arg}z$ with $-\pi <\mathrm{Arg}z\le \pi$. De Moivre's theorem states that $(\cos \theta +i\sin \theta {)}^n=\cos (n\theta )+i\sin (n\theta )$ for every integer $n$.

Since $\sqrt{3}+i=2e^{i\pi /6}$, the six roots are

Write $z=r{e}^{i\theta }$ with $-\pi <\theta \le \pi$. Its principal sixth root is $r^{1/6}{e}^{i\theta /6}$, whose complex argument lies in $(-\pi /6,\pi /6]$. Equality with $2e^{i\pi /6}$ forces $r=64$ and $\theta =\pi$, so $z=-64$.

For $z=x+iy$,Matching its modulus and complex argument with $2e^{i\pi /6}$ gives

The base $e^{5i\pi /2}$ is the complex number $i$. Principal exponentiation uses the principal logarithm of that base, so this is the same equation as in part iii: