# Absolute Value of Product/Proof 3

## Theorem

Let $x, y \in \R$ be real numbers.

Then:

$\size {x y} = \size x \size y$

where $\size x$ denotes the absolute value of $x$.

## Proof

 $\ds \size {x y}$ $=$ $\ds \sqrt {\paren {x y}^2}$ Definition 2 of Absolute Value $\ds$ $=$ $\ds \sqrt {x^2 y^2}$ $\ds$ $=$ $\ds \sqrt {x^2} \sqrt{y^2}$ $\ds$ $=$ $\ds \size x \cdot \size y$

$\blacksquare$