### Instruction: Compute $(2 \cos 20^\circ + 2i \sin 20^\circ)^6.$  Enter your answer in rectangular form.### Response: I recognize that this expression is the same as $2^6 e^{6i \cdot 20^\circ}$ by Euler's formula.
So I can simplify it to $64 e^{120^\circ i}$.
To convert it to rectangular form, I need to use the cosine and sine of 120 degrees.
I know that $\cos 120^\circ = -\frac{1}{2}$ and $\sin 120^\circ = \frac{\sqrt{3}}{2}$ by the unit circle.
So the expression becomes $64 \left(-\frac{1}{2} + \frac{\sqrt{3}}{2} i \right)$.
Distributing the 64, I get $-32 + 32 \sqrt{3} i$.