Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 1 / 1B / a

Squaring first avoids unnecessary expansion: $(2+2i{)}^2=8i$, and squaring again gives the fourth power $\boxed{z^4=-64}$. In the polar form of a complex number, $z=2\sqrt{2}{e}^{i\pi /4}$; multiplying its complex argument by four gives the same negative-real result.

Write a nonzero complex number as $w=r{e}^{i\theta }$. Matching modulus and complex argument gives $r^4=2\sqrt{2}$ and $4\theta =\pi /4+2\pi k$. Thus the four distinct fourth roots areThese angles differ by $\pi /2$, so the values are distinct. Every further integer $k$ repeats one of them; this is multiplication by the fourth roots of unity.