Question

What is the polar form of -2√3-6i ?

Answer 1

Any complex number $z$ can be written in polar/exponential form as

$z = |z| e^{i\arg(z)} = |z| (\cos(\arg(z)) + i \sin(\arg(z)))$

In the complex plane, the given complex number belongs to the third quadrant since both its real and imaginary parts are negative. This means the argument must be between π and 3π/2, so the first two options are eliminated.

The difference between the remaining two options is the coefficient $|z|$. We have

$z = -2\sqrt3 - 6i \implies |z| = \sqrt{(-2\sqrt3)^2 + (-6)^2} = 4\sqrt3$

so (D) (the last option) is the correct choice.