# What is the square root of -10 times the root of -40?

Sep 20, 2015

$\sqrt{- 10} \sqrt{- 40} = - 20$

#### Explanation:

$\sqrt{- 10} \sqrt{- 40} =$
$\left(\sqrt{- 10}\right) \left(\sqrt{- 40}\right) =$

You can't simply join the roots together, like $\sqrt{x} \sqrt{y} = \sqrt{x y}$, because that formula only works if $x$ and $y$ aren't both negative. You have to take the negative out of the root first and then multiply then, using the identity ${i}^{2} = - 1$ where $i$ is the imaginary unit, we continue like:

$\left(\sqrt{- 1} \sqrt{10}\right) \left(\sqrt{- 1} \sqrt{40}\right) =$
$\left(i \sqrt{10}\right) \left(i \sqrt{40}\right) =$
$\left({i}^{2} \sqrt{10} \sqrt{40}\right) =$
$- \sqrt{40 \cdot 10} =$
$- \sqrt{4 \cdot 100} =$
$- 20$

Sep 20, 2015

$\sqrt{- 10} \sqrt{- 40} = - 20$

#### Explanation:

Use these two complex number definitions/rules to simplify the expression: $\sqrt{- 1} = i$, and ${i}^{2} = {\sqrt{- 1}}^{2} = - 1$

$\sqrt{- 10} \sqrt{- 40} =$
$\sqrt{- 1 \cdot 10} \sqrt{- 1 \cdot 4 \cdot 10} =$
$\sqrt{- 1} \sqrt{10} \sqrt{- 1} \sqrt{4} \sqrt{10} =$
${\sqrt{- 1}}^{2} 2 {\sqrt{10}}^{2} =$
$- 1 \cdot 2 \cdot 10 = - 20$