Solution
= Solution
Write a nonzero <complex number> as $w=re^{i\theta}$. Matching <modulus> and <complex argument> gives $r^4=2\sqrt2$ and $4\theta=\pi/4+2\pi k$. Thus \b[the four distinct fourth roots are]
$$
\boxed{w_k=2^{3/8}\exp\!\left(i\left(\frac\pi{16}+\frac{k\pi}2\right)\right),\qquad k=0,1,2,3.}
$$
These 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>.