Solution
= Solution
The <complex number> $-1+i$ has <complex modulus> $\sqrt2$ and argument $3\pi/4$. A cube root has modulus $(\sqrt2)^{1/3}=2^{1/6}$, and its argument is one third of $3\pi/4+2k\pi$. Hence the three distinct roots are
$$
\boxed{z_k=2^{1/6}\exp\left[i\left(\frac\pi4+\frac{2k\pi}{3}\right)\right],
\qquad k=0,1,2.}
$$
Their displayed arguments are $\pi/4$, $11\pi/12$, and $19\pi/12$; the last is equivalently $-5\pi/12$ in the principal range.