Compute $(-1 + i \sqrt{3})^8.$  Enter the answer in rectangular form.
First, we convert $-1 + i \sqrt{3}$ to polar form, which gives us $2 \operatorname{cis} 120^\circ.$  Then by DeMoivre's Theorem,
\begin{align*}
(2 \operatorname{cis} 120^\circ)^8 &= 2^8 \operatorname{cis} 960^\circ \\
&= 256 \operatorname{cis} 240^\circ \\
&= 256 \left( -\frac{1}{2} - \frac{\sqrt{3}}{2} i \right) \\
&= \boxed{-128 - 128 \sqrt{3} i}.
\end{align*}