When we work with complex numbers, we usually see them in two main ways:

1. Rectangular form: This looks like $a + bi$, where $a$ and $b$ are real numbers, and $i$ represents the imaginary unit.

2. Polar form: This is written as $r(\cos \theta + i \sin \theta)$ or $re^{i\theta}$.

Both forms are useful, but they help us in different types of calculations.

1. Multiplying Complex Numbers:
When we multiply complex numbers, polar form makes it much easier.

For example, let’s say we have two complex numbers in polar form:
$z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$.

To find the product of these two, you use:
$z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$

This means you just multiply the sizes ($r_1$ and $r_2$) and add the angles ($\theta_1$ and $\theta_2$). It makes the math faster and simpler!

Example:
If $z_1 = 2e^{i\frac{\pi}{4}}$ and $z_2 = 3e^{i\frac{\pi}{2}}$, then the multiplication goes like this:
$z_1 z_2 = 2 \cdot 3 e^{i(\frac{\pi}{4} + \frac{\pi}{2})} = 6e^{i\frac{3\pi}{4}}$

2. Dividing Complex Numbers:
Polar form also helps us divide complex numbers easily. Using the same examples:
$\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$

So here, you take the sizes and divide them, then you subtract the angles. This method can be less complicated than using rectangular form.

3. Finding Roots of Complex Numbers:
When we want to find roots (like square roots) of complex numbers, polar form makes it clearer.

For example, to find the $n$th root of $z = re^{i\theta}$, you can calculate:
$\sqrt[n]{z} = r^{1/n} e^{i(\frac{\theta + 2k\pi}{n})} \quad (k = 0, 1, \ldots, n-1)$

In summary, both rectangular and polar forms have their own uses. But when it comes to multiplying, dividing, and finding roots of complex numbers, polar form makes everything much easier and faster!