Question Type 1: Finding powers of complex numbers Exercises

Evaluate $\left(\frac{1+i\sqrt{3}}{2}\right)^9$.

Compute $(\sqrt{3}+i)^6$ in the form $a+bi$.

Given $z=8\left(\cos \frac{3\pi}{4} + i\sin \frac{3\pi}{4}\right)$, compute $z^3$ in the form $a+bi$.

Let $z=2-2i$. Compute $z^8$ in the form $a+bi$.

Evaluate $\big(\cos \frac{\pi}{12} + i\sin \frac{\pi}{12}\big)^{24}$.

Evaluate $(1-i)^{12}$ in the form $a+bi$.

Let $z = 2\left(\cos \frac{2\pi}{7} + i\sin \frac{2\pi}{7}\right)$. Find $z^5$ in the form $r(\cos\theta + i\sin\theta)$ with $0\le \theta<2\pi$.

Evaluate $\left[3\left(\cos \frac{5\pi}{6} + i\sin \frac{5\pi}{6}\right)\right]^4$ in the form $a+bi$.

If $z=\cos\tfrac{\pi}{5}+i\sin\tfrac{\pi}{5}$, compute $\text{Re}(z^{10})$.

Evaluate $[5(\cos \tfrac{\pi}{9} + i\sin \tfrac{\pi}{9})]^3$ in the form $a+bi$.

Express $\dfrac{(2+2i)^{5}}{(\sqrt{3}-i)^{4}}$ in the form $a+bi$.

Find the least positive integer $n$ such that $\text{Im}\left[\left(\cos \frac{\pi}{9}+i\sin \frac{\pi}{9}\right)^n\right]=0$.