Why do books say $\sqrt{xy}=\sqrt x\sqrt y$ My question is simple. I'm teaching in a college (pre-calculus course) and I'm asking myself why high school books say $\sqrt{xy}=\sqrt x\sqrt y$. This is false (counterexample is $x=y=-1$).

How can I state this rule in a more general and correct manner?

Thanks
$\sqrt{x}:\mathbb{R}_{\ge0}\to\mathbb{R}_{\ge0}$ is defined as the unique $y\in\mathbb{R}_{\ge0}$ so that $y^2=x$ where $\mathbb{R}_{\ge0}=\left\{x\in\mathbb{R}:x\ge0\right\}$.

For this function, and this should be the function in your book, $\sqrt{xy}=\sqrt{x}\sqrt{y}$.



Unfortunately, after complex numbers are introduced, sometimes writers define an inverse of $z\to z^2$ on $\mathbb{C}$ minus some branch cut and call it $\sqrt{z}$. However, for this function, it is not the case that $\sqrt{zw}=\sqrt{z}\sqrt{w}$.
Corrected:

If $x$ and $y$ are allowed to be complex, your proposed counterexample is no such thing:

When $x = -1$ and $y = -1$,

$$\sqrt{xy} = \sqrt 1 = \pm 1$$

and

$$\sqrt x\sqrt y = \sqrt {-1} \sqrt {-1} = (\pm i) (\pm i) = \pm (-1) = \pm 1 $$ 

So the high-school text books are more correct than they seem to be. 



The above interprets $ \sqrt x $ as a multivalued function. If we take $\sqrt x$ as the principle value, the contention is, as you say, false.