# How do you simplify the square root of -6 times square root of -50?

Sep 16, 2015

#### Answer:

$\sqrt{- 6} \cdot \sqrt{- 50} = - 10 \sqrt{3}$

#### Explanation:

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

$- i \sqrt{- x}$ is also a square root of $x$.

If $a , b \ge 0$ then $\sqrt{a} \sqrt{b} = \sqrt{a b}$

The condition $a , b \ge 0$ is important. For example:

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

So:

$\sqrt{- 6} \cdot \sqrt{- 50} = i \sqrt{6} \cdot i \sqrt{50} = {i}^{2} \cdot \sqrt{6} \sqrt{50}$

$= - 1 \cdot \sqrt{300} = - \sqrt{{10}^{2} \cdot 3} = - \sqrt{{10}^{2}} \sqrt{3} = - 10 \sqrt{3}$