IB Mathematics Analysis and Approaches 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 $(\cos \frac{\pi }{12}+i\sin \frac{\pi }{12}{)}^{24}$.

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

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

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

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