Past exam of the mathematical methods I course of the University of Oxford / 2013 resit / 2 / Solution

Write $27i=27e^{i(\pi /2+2\pi k)}$. Every cube root has complex modulus $3$ and an angle one third of one of these complex arguments:Equivalently the three complex roots areThey lie on the circle of radius $3$ centred at the origin in the Argand plane, spaced by $2\pi /3$. Further integer values of $k$ repeat these three points. Cubing any of them multiplies its angle by three and cubes its complex modulus from $3$ to $27$, giving $27i$.