Answer
$\frac{1}{128}i$
Work Step by Step
DeMoivre’s Theorem:
For any complex number $z=r\left( \cos \theta +i\sin \theta \right)$, if n is a positive integer $\left( z>0 \right)$ then,
$\begin{align}
& {{z}^{n}}={{\left[ r\left( \cos \theta +i\sin \theta \right) \right]}^{n}} \\
& ={{r}^{n}}\left( \cos n\theta +i\sin n\theta \right)
\end{align}$
So the given complex number can be written as:
$\begin{align}
& {{\left[ \frac{1}{2}\left( \cos \frac{\pi }{14}+i\sin \frac{\pi }{14} \right) \right]}^{7}}={{\left( \frac{1}{2} \right)}^{7}}\left[ \cos 7\left( \frac{\pi }{14} \right)+i\sin 7\left( \frac{\pi }{14} \right) \right] \\
& =\frac{1}{128}\left( \cos \frac{\pi }{2}+i\sin \frac{\pi }{2} \right) \\
& =\frac{1}{128}\left\{ 0+i\left( 1 \right) \right\} \\
& =\frac{1}{128}i
\end{align}$
Hence, ${{\left[ \frac{1}{2}\left( \cos \frac{\pi }{14}+i\sin \frac{\pi }{14} \right) \right]}^{7}}=\frac{1}{128}i$.
