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

Compute $$1+i$$ to the 5th power and find its trigonometric form.

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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: $$$\displaystyle \begin{array}{rl} (1+i)^5=&\big( \sqrt{2}\cdot e^{i45^\circ}\big)^5 = (\sqrt{2})^5 \cdot (e^{i45^\circ})^5 \\ =& 4\sqrt{2} \cdot e^{i225^\circ}= 4\sqrt{2} \cdot [\cos(225^\circ)+i\cdot\sin(225^\circ)] \end{array}$$$


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

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