Question

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

Answer 1

Answer on Question #74589 – Math – Complex Analysis

Question

Apply De Moivre's Theorem to write $(\sqrt{3} + \mathrm{i})^{5}$ in the form $a + \mathrm{i}b$, with $a, b$ belongs to $R$.

Solution

Showing the polar form of $\sqrt{3} + \mathrm{i}$ is $2(\cos \pi / 6 + \mathrm{i} \sin \pi / 6)$. Thus we have

$$\begin{array}{l} (\sqrt{3} + \mathrm{i})^{5} = [2(\cos \pi / 6 + \mathrm{i} \sin \pi / 6)]^{5} \\ = 2^{5}(\cos \pi / 6 + \mathrm{i} \sin \pi / 6)^{5} \\ = 32(\cos 5\pi / 6 + \mathrm{i} \sin 5\pi / 6) \\ = 32(-\sqrt{3}/2 + 1/2 \mathrm{i}) \\ = -16\sqrt{3} + 16 \mathrm{i}. \end{array}$$

Answer: $-16\sqrt{3} + 16$ i

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