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 $(\tfrac12 e^{i\pi/7})^7$ 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 $(-\pi,\pi]$.