Absolute value is multiplicative

Statement

For all real numbers $a$ and $b$, $|a b| = |a| \, |b|$. Here $|\cdot|$ is the absolute value.

Remarks

Proof: check the four cases of the signs of $a$ and $b$. In particular $|a^n| = |a|^n$ for each natural number $n$, and $|a / b| = |a| / |b|$ for $b \neq 0$. The modulus of complex numbers is also multiplicative.

Explanations

L1 human

You can take the absolute value before or after a product, and you get the same result. For example, $|(-2) \cdot 3| = |-6| = 6$ and $|-2| \cdot |3| = 2 \cdot 3 = 6$.