Question Type 4: Applying multiplication and division to complex numbers in Euler form Exercises

Multiply $4e^{i\frac{\pi}{2}}$ by $e^{i\frac{\pi}{3}}$ and give the result in Euler form.

Multiply $2e^{i\frac{\pi}{6}} \times 3e^{-i\frac{\pi}{4}} \times 4e^{i\frac{\pi}{3}}$ and express your answer in Euler form.

Simplify $\displaystyle\frac{3e^{i\pi/3}}{6e^{-i\pi/4}} \times 2e^{i\pi/12}$ and express the answer in Euler form.

Simplify $\frac{6e^{i\frac{3\pi}{2}} \times 4e^{-i\frac{\pi}{6}}}{2e^{i\frac{\pi}{3}}}$ and express your answer in $a+bi$ form.

Compute $\displaystyle\frac{5e^{i\pi/2}}{8e^{i\pi/4}}$ and express the answer in Euler form.

Compute $\displaystyle\frac{5e^{i7\pi/4}}{4e^{i3\pi/2}}$ and express the argument in the principal range $(-\pi,\pi]$.

Compute $\displaystyle\frac{8e^{-i\pi/6}}{2e^{-i\pi/2}}$ in Euler form.

Multiply $5\mathrm{e}^{\mathrm{i}\frac{2\pi}{3}} \times 3\mathrm{e}^{\mathrm{i}\frac{5\pi}{6}}$ and express the result in the form $a+b\mathrm{i}$, where $a, b \in \mathbb{R}$.

Compute $\displaystyle\frac{6e^{-i\pi/3}}{3e^{i\pi/4}}$ and express the result in Euler form.

Multiply $2e^{i\frac{\pi}{6}}$ by $3e^{i\frac{\pi}{3}}$ and express the result in Euler form.

Compute $\displaystyle\frac{7e^{i4\pi/5}}{5e^{i9\pi/10}}$ and express the result in Euler form.

Multiply $7e^{i5\pi/4}$ by $2e^{i2\pi/3}$ and express the result in Euler form $re^{i\theta}$, where $-\pi < \theta \le \pi$.