
\begin{enumerate}
\item $\begin{aligned}[t]
(1+\i)^9 & = \left[ \sqrt2\left( \cos\dfrac{\pi}{4}+\i\sin\dfrac{\pi}{4}\right) \right]^9\\
& = 16\sqrt2\left( \cos\dfrac{9\pi}{4} + \i\sin\dfrac{9\pi}{4} \right)\\
& = 16\sqrt2\left( \cos\dfrac{\pi}{4} + \i\sin\dfrac{\pi}{4} \right)\\
& = 16\sqrt2\left(\dfrac{\sqrt2}{2}+\i\dfrac{\sqrt2}{2}\right)\\
\Aboxed{ (1+\i)^9 & = 16 + 16\i}
\end{aligned}$

\item $\begin{aligned}[t]
\big(\sqrt3-\i\big)^{6} & = \left[ 2\left(\dfrac{\sqrt3}{2}-\dfrac{1}{2}\i\right) \right]^6\\
& = \left[ 2\left( \cos\left(-\dfrac{\pi}{6}\right) + \i\sin\left(-\dfrac{\pi}{6}\right) \right) \right]^6\\
& = 2^6\big( \cos(-\pi) + \i\sin(-\pi) \big)\\
\Aboxed{ \big(\sqrt3-\i\big)^{6} & = -64 }
\end{aligned}$

\item $\begin{aligned}[t]
\big(1+\i\sqrt3\big)^9 & = \left[ 2\left( \dfrac{1}{2}+\dfrac{\sqrt3}{2}\i \right) \right]^9\\
& = \left[ 2\left( \cos\dfrac{\pi}{3}\i\sin+\dfrac{\pi}{3} \right) \right]^9\\
& =2^9\left( \cos\dfrac{9\pi}{3}\i\sin+\dfrac{9\pi}{3} \right)\\
& = 512\big( \cos(3\pi) + \i\sin(3\pi) \big)\\
& = 512\big( \cos(\pi) + \i\sin(\pi) \big)\\
\Aboxed{\big(1+\i\sqrt3\big)^9& = -512}
\end{aligned}$
\end{enumerate}
