= Solution
The <polar form> is $z=e^{3\pi i/4}$. Interpreting the fractional <complex exponentiation> as all its values, a candidate $w$ satisfies $w^3=z^2=-i$. The three <roots of a complex number> are
$$
w_0=i,\qquad w_1=-\frac{\sqrt3}{2}-\frac i2,\qquad w_2=\frac{\sqrt3}{2}-\frac i2.
$$
They have <arguments> $\pi/2$, $7\pi/6$ and $11\pi/6$, respectively. Since all have <modulus> one, their squared distances from $z$ in the <complex plane> are $|w-z|^2=2-2\cos(\arg w-3\pi/4)$, giving
$$
|w_0-z|^2=2-\sqrt2,\quad
|w_1-z|^2=2-\frac{\sqrt6-\sqrt2}{2},\quad
|w_2-z|^2=2+\frac{\sqrt6+\sqrt2}{2}.
$$
The last is strictly largest, so the required <root of a complex number> is
$$
\boxed{w=\frac{\sqrt3-i}{2}.}
$$
Restricting <complex exponentiation> to its principal value would give only $i$; the request to choose among roots requires all three values.
\Image[/past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2021-resit-complex-powers.png]
{title=The three values of the fractional complex power and their distances from z}
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