Exponents and absolute value w/ complex numbers

Question

A simple stupid question that I should already know the answer to:

Is this true: $\left|z^5\right| = \left|z\right|^5$ where z is a complex number with a non-zero imaginary component.

Thanks in advance.

Answer 1

Write in complex form

$z = re^{i\theta} \implies |z|^n = r^n|e^{i\theta}|^n = r^n\cdot1 = r^n$

$|z^n| = |r^ne^{in\theta}| = r^n|e^{in\theta}| = r^n\cdot 1 = r^n$

Answer 2

Yes, it is true.

Note that for any two complex numbers $z$ and $w$, we have

$$|zw|=|z||w|.$$

With $z=w$ we get $|z^2|=|z|^2$, and using induction you get $$|z^n|=|z|^n$$ for any natural number $n$.

Answer 3

This follows from the identity $$|zw|=|z||w|,$$ which may be verified by writing both the involved numbers in either rectangular or polar form, computing both sides separately and noting that they are equal.

By taking $z=w$ and using the identity repeatedly, you have that $|z^n|=|z|^n$ for any positive integer $n.$