$\textbf{Problem}$ (3pt). Find all the value(s) of the given expression $$ i^{\sqrt{3}}.$$
\begin{align*}\
i^{\sqrt{3}} &= e^{\ln(i)^{\sqrt{3}}} \\
&= e^{\sqrt{3} \ln(i)} \\
&= e^{\sqrt{3}(\ln \lvert{i}\rvert + i \arg{i})} \qquad \quad \ln(i) = \ln \lvert i \rvert + i \arg{i} \text{ since } i \in \mathbb{C}.\\
&= e^{\sqrt{3}(\ln{1} \, + \, i \left(\frac{\pi}{2} + 2\pi k \right))} \qquad \, k \in \mathbb{Z} \\
&= e^{\sqrt{3}(i \left(\frac{\pi}{2} + 2\pi k \right))} \\
&= \cos{\sqrt{3} \left(\frac{\pi}{2} + 2 \pi k \right)} + i \sin{\sqrt{3} \left(\frac{\pi}{2} + 2 \pi k \right)}
\end{align*}
