From Complex number to polar I have to calculate $(3 + 3i)^{829}$
What I did:
$$|3 + 3i| = \sqrt{3^2 + 3^2} = \sqrt{9 + 9}=\sqrt{18}$$
The argument of $3 + 3i$ is  $45º$.

The key is in the title: you need to write the number in polar form:
$$x+\mathrm i y = r(\cos\theta+\mathrm i \sin\theta)$$
where $r$ is the modulus of $x+\mathrm i y$ and $\theta$ is the argument.
In your case, you have $x + \mathrm i y = 3 + 3\mathrm i$.
The modulus is given by $\left|3+3\mathrm i\right|=\sqrt{3^2+3^2}=3\sqrt{2}$.
Since $3+3\mathrm i$ lies on the line $y=x$, and is in the upper-right quadrant. The argument is given by $\arg(3+3\mathrm i)=\frac{1}{4}\pi$.
That means that $r=3\sqrt{2}$ and $\theta=\frac{1}{4}\pi$, and hence
$$3+3\mathrm i = 3\sqrt{2}\left(\cos(\tfrac{1}{4}\pi)+\mathrm i \sin(\tfrac{1}{4}\pi)\right)$$
De Moivre's Theorem tells is that $(\cos \theta + \mathrm i \sin \theta)^n = \cos (n\theta) + \mathrm i \sin (n\theta)$.
It follows that
\begin{eqnarray*}
(3+3\mathrm i)^{829} &=& (3\sqrt 2)^{829}\left(\cos(\tfrac{829}{4}\pi)+\mathrm i \sin(\tfrac{829}{4}\pi)\right)
\end{eqnarray*}
The sine and cosine functions are $2\pi$-periodic, i.e. they repeat every $2\pi$ and so $\sin(x+2\pi)\equiv \sin x$ just like $\cos(x+2\pi) \equiv \cos x$ for all real $x$. We can subtract multiples of $2\pi=\frac{8}{4}\pi$ from $\frac{829}{4}\pi$ without changing the values of sine and cosine.
Since $829 = 4 \times 206 + 5$ it follows that $\frac{829}{4} = 206 + \frac{5}{4}$ and hence $\frac{829}{4}\pi = 206\pi + \frac{5}{4}\pi$.
Given that sine and cosine are $2\pi$ periodic:
\begin{eqnarray*}
\cos(\tfrac{829}{4}\pi) &=& \cos(206\pi + \tfrac{5}{4}\pi) \\ \\
&=& \cos(\tfrac{5}{4}\pi) \\ \\
\sin(\tfrac{829}{4}\pi) &=& \sin(206\pi + \tfrac{5}{4}\pi) \\ \\
&=& \sin(\tfrac{5}{4}\pi)
\end{eqnarray*}
It follows that 
\begin{eqnarray*}
(3\sqrt 2)^{829}\left(\cos(\tfrac{829}{4}\pi)+\mathrm i \sin(\tfrac{829}{4}\pi)\right) &=& (3\sqrt 2)^{829}\left(\cos(\tfrac{5}{4}\pi)+\mathrm i \sin(\tfrac{5}{4}\pi)\right) \\ \\
&=& (3\sqrt 2)^{829}\left(-\tfrac{1}{\sqrt 2}-\tfrac{1}{\sqrt 2}\,\mathrm i \right) \\ \\
&=& -3^{829}\times 2^{414} \times (1+\mathrm i)
\end{eqnarray*}
This number is massive and shouldn't be written down.
