= Solution
The right-hand side has <modulus> two and argument $\pi/3$. Its fifth roots therefore have
$$
\boxed{z_k=2^{1/5}\exp\left[i\left(\frac\pi{15}+\frac{2\pi k}{5}\right)\right],\qquad k=0,1,2,3,4.}
$$
They are the vertices of a regular pentagon on the circle of radius $2^{1/5}$, beginning at angle $\pi/15$. These five roots are distinct, and a degree-five <polynomial> has no further <complex roots>.
\Image[/past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2022-complex-root-loci.png]
{title=The fifth roots, the discrete ray of values of a complex power, and the unit-circle locus with its excluded point}
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