How to simplify ${(1+2i)}^6$? How to simplify ${(1+2i)}^6$ using De Moivre's formula?
I have found that $r=\sqrt 5$ and $\tan x=2$ but I can't find the exact value of $x$.

$$|1+2i|=\sqrt5\;,\;\;\arg(1+2i)=\arctan 2\implies (1+2i)^6=5^3e^{6\arctan 2\cdot i} $$
and now:
$$\begin{cases}&5^3=125\\{}\\
&e^{6\arctan2\cdot i}=0.936+0.352\,i\end{cases}\implies(1+2i)^6=125(0.936+0.352\,i)=117+44i$$
But this looks weird and, anyway, is way simpler first calculating the third power and then squaring...
