Calculate

$$
\begin{aligned}
& \text { a) }(1+i)^{n} ; \text { b) }(1+i \sqrt{3})^{n} ; \text { c) }\left(\frac{1+i \sqrt{3}}{1-i}\right)^{20} \text {; d) }\left(1-\frac{\sqrt{3}-i}{2}\right)^{20} \text {; e) }(1+\cos \varphi+i \sin \varphi)^{n} \text {; f) } \\
& (\sqrt{3}+i)^{n} ; \text { g) }\left(\frac{\cos \varphi+i \sin \varphi}{\cos \psi+i \sin \psi}\right)^{n}
\end{aligned}
$$
Represent each of the numbers in trigonometric form.

## Answer

a) $2^{n / 2}\left(\cos n \pi / 4 + i \sin n \pi / 4\right)$; b) $2^{n}\left(\cos n \pi / 3 + i \sin n \pi / 3\right)$; c) $-512(1 + i \sqrt{3})$; d) 

$$
-\frac{(\sqrt{3}-1)^{20}}{2^{10}}\left(\cos \frac{\pi}{9} + i \sin \frac{\pi}{9}\right)
$$

e) $2^{n} \cos ^{n} \varphi\left(\cos \frac{n \varphi}{2} + i \sin \frac{n \varphi}{2}\right)$; f) $2^{n}\left(\cos n \pi / 6 + i \sin n \pi / 6\right)$; g) $(\cos n(\varphi - \psi) + i \sin n(\varphi - \psi))$.