Complex 1304: polar form, arithmetic and conditions to be real/purely imaginary (basic)

In this question, we will convert complex numbers from cartesian form $a+bi$ to exponential form $re^{i\theta}$ and vice versa.

We also perform multiplication, division, exponentiation and conjugation to complex numbers in polar form.

We also work with conditions for a complex number in polar form to be real/purely imaginary.

Question types

convert cartesian form $a+bi$ to exponential form $re^{i\theta}$
convert exponential form to cartesian form
convert cartesian form to exponential form, where the values of $a,b,\theta$ are special trigonometric ratios/angles
convert exponential form to cartesian form, where the values of $a,b,\theta$ are special trigonometric ratios/angles
complex arithmetic in polar form, where the expression to evaluate is one of $\displaystyle \frac{z_1z_2^*}{z_3^n}, \frac{z_1z_2^n}{z_3^*}, \frac{ \left (z_1z_2^n \right)^*}{z_3}, \left ( \frac{z_1^nz_2}{z_3} \right)^*, \frac{\left(z_1^*z_2\right )^n}{z_3}$
conditions for $\left ( e^{i \theta} \right)^n$ $(e^{i θ})^{n}$ to be
- real
- real and positive
- real and negative
- purely imaginary