Simplify \[\left( \frac{3 + i \sqrt{3}}{2} \right)^8 + \left( \frac{3 - i \sqrt{3}}{2} \right)^8.\]
We write the complex numbers in exponential form:
(2*e^(pi*i/8))^8 + (2*e^(-pi*i/8))^8 = 256*e^(pi*i) + 256*e^(-pi*i) = -512.
