= Solution
Write $27=27e^{2\pi i n}$. The three <complex roots> are
$$
\boxed{z=3,\qquad z=-\frac32+\frac{3\sqrt3}{2}i,\qquad
z=-\frac32-\frac{3\sqrt3}{2}i.}
$$
On the <Argand diagram>, these are three equally spaced points on the circle $|z|=3$, with <arguments> $0,2\pi/3,4\pi/3$. They are isolated points, rather than the whole circle.
\Image[/past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2016-resit-loci-and-ode.png]
{title=Complex root and parabola loci, a first-order differential-equation solution, and rolling-point trajectories}
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