Find the value of complex expression $\left(\frac{\sqrt{3}+i}{2}\right)^{69}$ Find the value of


  $$\left(\dfrac{\sqrt{3}+i}{2}\right)^{69}.\DeclareMathOperator{\cis}{cis}$$


I tried to solve this complex expression by converting it into polar form.
I expressed it in polar form $r\cis(t)$ from rectangular form $x+iy$ where $\cis(t) = \cos(t) + i\sin(t)$.
But I am unable to solve further due to the exponent of 69!
$$\left(\dfrac{\sqrt{3}+i}{2}\right)^{69}=\left(\dfrac{\sqrt{3}}{2}+\dfrac{1}{2}i\right)^{69}=(\cos\dfrac{\pi}{6}+i\sin\dfrac{\pi}{6})^{69}=\cos\dfrac{69\pi}{6}+i\sin\dfrac{69\pi}{6}=-i$$
by De Moivre's formula.
$$\frac{\sqrt3+i}{2}=e^{i\frac{\pi}{6}}$$

$$\big(\frac{\sqrt3+i}{2}\big)^{69}=e^{i\frac{\pi}{6}\cdot69}=e^{i(11\pi+\frac{\pi}{2})}=e^{i(12\pi-\frac{\pi}{2})}=e^{-i\frac{\pi}{2}}=-i$$

I used the fact that $e^{i(2n\pi)}=1$ where $n\in Z$