How do you compute an expression containing complex numbers with large powers?

Question

$$(\frac{-\sqrt{3}}{2}+\frac{1}{2}i)^{123}=i$$

$$(\frac{-3}{\sqrt{2}}+\frac{-3}{\sqrt{2}}i)^{11}=\frac{3^{11}}{\sqrt{2}}-\frac{3^{11}}{\sqrt{2}}i$$

So I have these equations with the answers already. I was wondering, how do you get them? What patterns should I look?

Answer 1

One way is to express them in polar form, so $\frac{-3}{\sqrt{2}}+\frac{-3}{\sqrt{2}}i=3e^{i\frac {5\pi}4}$. Then you can use the law for the power of a power $$(\frac{-3}{\sqrt{2}}+\frac{-3}{\sqrt{2}}i)^{11}=(3e^{i\frac {5\pi}4})^{11}=3^{11}e^{i\frac{55\pi}4}=\frac{3^{11}}{\sqrt{2}}-\frac{3^{11}}{\sqrt{2}}i$$

Answer 2

In general with response to @Ross Millikan's answer, given an complex number $z=a+bi$ to compute $z^n$ write $z=re^{i\theta}=|z|e^{i\theta}$ i.e in polar form then $z^n = r^ne^{in\theta}=|z|^ne^{in\theta}$.