IB Mathematics Applications & Interpretation Question Type 6: Applying Powers to Complex Numbers in Euler Form Exercises

Calculate $(3e^{-i\pi /4}{)}^4$ and write the answer in $a+bi$ form.

Compute $(2e^{i\frac{\pi }{6}}{)}^5$ and give your answer in Euler form.

Let $z=5e^{i\pi /5}$. Find $z^8$ and write your answer in modulus–argument form with the argument in $(-\pi ,\pi ]$.

Find $(3{\text{e}}^{\text{i}\frac{2\pi }{5}}{)}^5$, giving your answer in Euler form.

Calculate $(4e^{-i\pi /3}{)}^3$ and express the result in $a+bi$ form.

Calculate $(2e^{i\pi /3}{)}^{-2}$ and express the result in $a+bi$ form.

For $w=7e^{i\frac{2\pi }{3}}$, compute $w^{-3}$ and state its principal argument in the interval $(-\pi ,\pi ]$.

Find $(\sqrt{2}{e}^{i3\pi /4}{)}^2$ and express the answer in Euler form.

Calculate $(5e^{i\frac{\pi }{2}}{)}^3$ and express the result in the form $a+bi$, where $a,b\in \mathbb{R}$.

Evaluate $(\frac{1}{2}{e}^{i\pi /7}{)}^7$ and express your answer in $a+bi$ form.

Evaluate ${\left(e^{i\frac{\pi }{3}}\right)}^6$ and simplify fully.

Find the modulus and argument of $(6{\mathrm{e}}^{\mathrm{i}3\pi /4}{)}^3$, giving the argument in $(-\pi ,\pi ]$.