Aus Online Mathematik Brückenkurs 2

First, we write the number \displaystyle \frac{1}{2}+i\frac{\sqrt{3}}{2} in polar form.

Thus,

\displaystyle \frac{1}{2}+i\frac{\sqrt{3}}{2}=1\centerdot \left( \cos \frac{\pi }{3}+i\sin \frac{\pi }{3} \right)

and de Moivre's formula gives

\displaystyle \begin{align} & \left( \frac{1}{2}+i\frac{\sqrt{3}}{2} \right)^{12}=1^{12}\centerdot \left( \cos 12\centerdot \frac{\pi }{3}+i\sin 12\centerdot \frac{\pi }{3} \right) \\ & =1\centerdot \left( \cos 4\pi +i\sin 4\pi \right) \\ & =1\centerdot \left( 1+i\centerdot 0 \right)=1 \\ \end{align}