Question Type 1: Finding powers of complex numbers Bootcamps

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

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

Evaluate $\Big(\tfrac{1+i\sqrt{3}}{2}\Big)^9$.

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

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

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

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

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

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

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

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

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