# How do you simplify  2sqrt(-27) • sqrt(-3)?

Mar 2, 2018

$2 \sqrt{- 27} \sqrt{- 3} = - 18$

#### Explanation:

It is interesting to note what you should not do.

Note that:

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

if at least one of $a$ and $b$ is non-negative.

In our example we have $a = - 27 < 0$ and $b = - 3 < 0$ and we might erroneously deduce:

$\textcolor{red}{\cancel{\textcolor{b l a c k}{2 \sqrt{- 27} \sqrt{- 3} = 2 \sqrt{\left(- 27\right) \left(- 3\right)} = 2 \sqrt{81} = 2 \cdot 9 = 18}}}$

Instead, note that if $a < 0$ then $\sqrt{a} = i \sqrt{- a}$

So we have:

$2 \sqrt{- 27} \sqrt{- 3} = 2 i \sqrt{27} i \sqrt{3} = 2 {i}^{2} \sqrt{27} \sqrt{3} = - 2 \sqrt{81} = - 2 \cdot 9 = - 18$