# Why is sqrt(-2) sqrt(-3) != sqrt((-2) * (-3)) ?

Jan 9, 2017

The "rule" $\sqrt{a b} = \sqrt{a} \sqrt{b}$ does not hold all of the time - especially when it comes to negative or complex numbers.

#### Explanation:

Here's another example:

$1 = \sqrt{1} = \sqrt{\left(- 1\right) \cdot \left(- 1\right)} \ne \sqrt{- 1} \cdot \sqrt{- 1} = - 1$

This kind of thing happens because every non-zero number has two square roots and the one we mean when we write $\sqrt{\ldots}$ can be a slightly arbitrary choice.

So long as you stick to $a , b \ge 0$ then the rule holds:

$\sqrt{a b} = \sqrt{a} \sqrt{b}$

When you get to deal with complex numbers more fully (in precalculus?) then it may become clearer.