Express the following complex numbers in standard form $$\left(\frac{\sqrt 3}{2}+\frac{i}{2}\right)^{25}$$
I know that you have to put it in the form $\cos\theta+i\sin \theta$ but I'm not sure how to go about it. 

Applying De-moivre's theorem,   $(\cos\theta+i\sin \theta)^n=\cos n\theta+i\sin n\theta$
hence, $$\left(\frac{\sqrt 3}{2}+\frac{i}{2}\right)^{25}$$
$$=\left(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\right)^{25}$$
$$=\cos\frac{25\pi}{6}+i\sin\frac{25\pi}{6}$$
$$=\cos\left(4\pi+\frac{\pi}{6}\right)+i\sin\left(4\pi+\frac{\pi}{6}\right)$$
$$=\cos\left(\frac{\pi}{6}\right)+i\sin\left(\frac{\pi}{6}\right)$$
$$=\color{red}{\frac{\sqrt 3}{2}+\frac{i}{2}}$$
