# How do you simplify sqrt(-3)sqrt(-12)?

Oct 30, 2017

$\sqrt{- 3} \sqrt{- 12} = - 6$

#### Explanation:

What is interesting about this example is that it is a counterexample to the often quoted "identity":

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

In fact this identity only holds if at least one of $a$ or $b$ is non-negative.

By convention, the principal square root of a negative number $n$ is given by:

$\sqrt{n} = i \sqrt{- n}$

So:

$\sqrt{- 3} \sqrt{- 12} = i \sqrt{3} \cdot i \sqrt{12} = {i}^{2} \sqrt{3} \sqrt{12} = - 1 \cdot \sqrt{3 \cdot 12} = - \sqrt{36} = - 6$