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

Development:

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: $\alpha =\arctan ({\displaystyle \frac{1}{1}})\Rightarrow \alpha ={45}^{\circ }$

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

Solution:

$(1+i{)}^5=4\sqrt{2}\cdot [\cos ({225}^{\circ })+i\cdot \sin ({225}^{\circ })]$.

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