Exponentiation and roots of complex numbers in trigonometric form (Moivre's formula)

First we convert $1+i$ to its trigonometric form:

We compute its norm: $|1+i|=\sqrt{1^2+1^2}=\sqrt{2}$

And now its argument: $\alpha=\arctan\big( \dfrac{1}{1}\big) \Rightarrow \alpha=45^\circ$

Therefore we can write it as: $$1+i=\sqrt{2}\cdot [\cos(45^\circ)+i\cdot\sin(45^\circ)] =\sqrt{2}\cdot e^{i45^\circ}$$ Now we compute the power: $$\displaystyle \begin{array}{rl} (1+i)^5=&\big( \sqrt{2}\cdot e^{i45^\circ}\big)^5 = (\sqrt{2})^5 \cdot (e^{i45^\circ})^5 \\ =& 4\sqrt{2} \cdot e^{i225^\circ}= 4\sqrt{2} \cdot [\cos(225^\circ)+i\cdot\sin(225^\circ)] \end{array}$$