### Instruction: Compute $(-1 + i \sqrt{3})^8.$  Enter the answer in rectangular form.### Response: I recognize that the complex number $-1 + i \sqrt{3}$ has magnitude $2$ and angle $\frac{2 \pi}{3}$ in polar coordinates.
So I can write it as $2 e^{i \frac{2 \pi}{3}}.$
Using the property of exponents, I can raise both sides to the eighth power and get $(-1 + i \sqrt{3})^8 = (2 e^{i \frac{2 \pi}{3}})^8 = 2^8 e^{i \frac{16 \pi}{3}}.$
Simplifying, I get $(-1 + i \sqrt{3})^8 = 256 e^{i \frac{4 \pi}{3}}.$
To convert back to rectangular form, I use the identity $e^{i \theta} = \cos \theta + i \sin \theta.$
So I get $(-1 + i \sqrt{3})^8 = 256 (\cos \frac{4 \pi}{3} + i \sin \frac{4 \pi}{3}).$
Using the unit circle, I know that $\cos \frac{4 \pi}{3} = -\frac{1}{2}$ and $\sin \frac{4 \pi}{3} = -\frac{\sqrt{3}}{2}.$
So I get $(-1 + i \sqrt{3})^8 = 256 (-\frac{1}{2} - i \frac{\sqrt{3}}{2}).$
Multiplying by $256$, I get $(-1 + i \sqrt{3})^8 = -128 - i 128 \sqrt{3}.$