A complex number raised to a power.

Question

$(4\sqrt{3} +4i)^3 = $ ?

I had $148.1028 + 4.0611i$

Answer 1

$$4^3(\sqrt3^3+i3\sqrt3^2-3\sqrt3-i)=64(i8).$$

Answer 2

Hint:

It's $\left[8\left(\dfrac{\sqrt3+i}2\right)\right]^3$

Answer 3

Factoring gives

$$(4\sqrt{3}+4i)^3=8^3\left(\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}i\right)^3\\$$ $$=8^3\left(e^{i\frac\pi6}\right)^3\\$$ $$=8^3e^{i\frac\pi2}\\$$ $$=512i$$