Question

Apply De Moivre’s theorem to write (√3+i)^5 in the form a + ib , with
a, b∈ R

Answer 1

Answer on Question #78434 – Math – Complex Analysis

Question

Apply De Moivre’s theorem to write $(\sqrt{3} + \mathrm{i})^{\wedge}5$ in the form $a + ib$, with $a, b \in \mathbb{R}$

Solution

$$z = \sqrt {3} + i = 2 \left(\cos \frac {\pi}{6} + i \sin \frac {\pi}{6}\right).$$ $$z ^ {5} = 2 ^ {5} \left(\cos \frac {5 \pi}{6} + i \sin \frac {5 \pi}{6}\right) = 3 2 \left(- \frac {\sqrt {3}}{2} + \frac {1}{2} i\right) = - 1 6 \sqrt {3} + 1 6 i.$$

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