Question type 7: Multiplying and dividing complex numbers in Euler form Exercises

Compute $z = 4e^{i\frac{\beta}{2}} \times 2e^{i\pi}$ where $\beta = \pi$. Express $z$ in Euler form with principal argument and in Cartesian form $a+bi$, and state its modulus and principal argument.

Find $z = 3e^{i\frac{\pi}{3}} \times 2e^{-i\frac{\pi}{4}}$. Give the result in Euler form with principal argument and in Cartesian form.

Compute $z = \dfrac{6e^{i\frac{3\pi}{4}}}{2e^{i\frac{\pi}{6}}}$. Express $z$ in Euler form with the principal argument and in Cartesian form $a+bi$.

Evaluate $w = \dfrac{10e^{i\frac{9\pi}{4}}}{5e^{i\frac{3\pi}{2}}}$ and express the result in Euler form with principal argument and in Cartesian form.

Find the real $k>0$ such that $(ke^{i\frac{\pi}{3}})\big(2e^{i\frac{\pi}{6}}\big) = 10e^{i\frac{\pi}{2}}$. [3 marks]

Given $z = 4e^{-i\frac{2\pi}{3}}$, find $\dfrac{1}{z}$ in exponential form and Cartesian form.

Evaluate $z = \dfrac{8e^{-i\frac{5\pi}{4}}}{3e^{i\frac{\pi}{12}}}$ and give $z$ in exponential form with principal argument and in Cartesian form.

Compute $z = \dfrac{5e^{i\frac{5\pi}{3}}}{2e^{i\frac{7\pi}{6}}}$ and express your answer in Euler form with principal argument and as $a+bi$.

Find $q = \dfrac{7e^{i\frac{11\pi}{6}}}{14e^{-i\frac{\pi}{3}}}$. Give $q$ in exponential form with principal argument and in Cartesian form.

Compute $z = \left(5e^{i\frac{7\pi}{6}}\right) \times \left(\frac{1}{2}e^{-i\frac{5\pi}{6}}\right)$. Express $z$ in Euler and Cartesian forms.

Let $z = e^{i\frac{11\pi}{8}}$ and $w = e^{-i\frac{5\pi}{12}}$. Compute $\dfrac{z^3}{w^2}$ in Euler form with the principal argument.

Compute $z = 2e^{i\frac{\pi}{6}} \cdot 3e^{i\frac{5\pi}{6}}$ and give its Cartesian coordinates for plotting on an Argand diagram.