NT Warm-up

$\begin{aligned} z & = \left(\frac 12 + \frac {\sqrt 3}2i \right)^{2015} \\ & = \left(\color{#3D99F6} \cos \frac \pi 3 + i\sin \frac \pi 3 \right)^{2015} & \small \color{#3D99F6} \text{By Euler formula} \\ & = \left(\color{#3D99F6} e^{\frac \pi 3 i} \right)^{2015} = e^{671\frac 23i} & \small \color{#3D99F6} \text{Since }e^{2\pi} = 1 \\ & = e^{\frac 53\pi i} \\ & = \cos \frac 53 \pi + i \sin \frac 53 \pi \\ & = \boxed {\dfrac 12 - \dfrac {\sqrt 3}2 i} \end{aligned}$

$\left(\dfrac{1}{2} + \dfrac{\sqrt{3}}{2}i\right)^{2015}$

$=\left(\cos\dfrac{\pi}3+i\sin\dfrac{\pi}3\right)^{2015}$

$=\cos\dfrac{2015\pi}3+i\sin\dfrac{2015\pi}3$

$=\dfrac{1}{2} - \dfrac{\sqrt{3}}{2}i$