Does the complex number $3 + i\sqrt3 $ have a purely imaginary power? I'm trying to find if this affirmation is true or false: 

As $z= 3 + i\sqrt3 $ where $i$ is the imaginary unit

I would know if ∀$n$ $\in \mathbb{N} \neq 0$, $z^{3n}$ is a pure imaginary number.

I tried some weird calculations but nothing work... Could you help me?
If you are not familiar with polar coordinates yet, use induction. 

Let $z = 3 + \sqrt{3}i$. We claim that $z^{3(2n+1)}$ is purely imaginary for $n \in \mathbb{N}_0$. 

For $n=0$, we have
$$z^3 = 3^3 + 3 \cdot 3^2 \cdot \sqrt{3} i - 3 \cdot 3 \cdot 3 - 3 \sqrt{3}i,$$
which is clearly purely imaginary. Assume $z^{3(2n+1)}$ is purely imaginary for some $n \in \mathbb{N}_0$. Then,

$$z^{3(2n+3)} = z^{3(2n+1)+6} = z^{3(2n+1)} z^6.$$

Note that $z^6 =z^3 \cdot z^3$, which must be purely real since $z^3$ is purely imaginary. Therefore, $z^{3(2n+3)}$ is purely imaginary.

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In polar coordinates, 

$$z = 3 + \sqrt{3} i = \sqrt{12} e^{i \pi/6}.$$

Recall that $e^{i\pi (n + 1/2)}$ is purely imaginary for all $n \in \mathbb{Z}$.
$(3+i\sqrt 3)^n=(2\sqrt 3)^n (\sqrt 3 /2 +i/2))^n=(2\sqrt 3)^n (\cos 30^o+i\sin 30^o)^n.\quad$ Use DeMoivre's Theorem :$ (\cos A +i\sin A)^n=\cos (n A)+i\sin (n A).$ Observe that $\cos (30n)^o=0 $iff $n/3$ is an odd integer.