# How do you simplify sqrt(-9)?

Feb 19, 2016

$\sqrt{- 9} = 3 i$

#### Explanation:

If $x < 0$ then $\sqrt{x} = \sqrt{- x} i$, where $i$ is the imaginary unit.

$i$ has the property that ${i}^{2} = - 1$

In our example, we find:

$\sqrt{- 9} = \sqrt{9} i = 3 i$

Note that special care is required when dealing with square roots of negative numbers. In particular $\sqrt{a b} \ne \sqrt{a} \sqrt{b}$ if both $a < 0$ and $b < 0$.

For example:

$1 = \sqrt{1} = \sqrt{- 1 \cdot - 1} \ne \sqrt{- 1} \cdot \sqrt{- 1} = - 1$