Answer
The rectangular form of the expression is $2+2i\sqrt{3}$.
Work Step by Step
Consider the following expression,
${{\left[ \sqrt{2}\left( \cos 15{}^\circ +i\sin 15{}^\circ \right) \right]}^{4}}$
Use the formula ${{\left( \cos \left( x \right)+i\sin \left( x \right) \right)}^{n}}=\cos \left( nx \right)+i\sin \left( nx \right)$ to solve the expression ${{\left[ \sqrt{2}\left( \cos 15{}^\circ +i\sin 15{}^\circ \right) \right]}^{4}}$ ,
$\begin{align}
& {{\left[ \sqrt{2}\left( \cos 15{}^\circ +i\sin 15{}^\circ \right) \right]}^{4}}={{\left( \sqrt{2} \right)}^{4}}\left[ \cos 4\left( 15{}^\circ \right)+i\sin \left( 15{}^\circ \right) \right] \\
& =4\left( \cos 60{}^\circ +i\sin 60{}^\circ \right) \\
& =4\left( \frac{1}{2}+\frac{\sqrt{3}}{2}i \right) \\
& =2+2i\sqrt{3}
\end{align}$
Hence, the rectangular form of the expression is $2+2i\sqrt{3}$.
