# What is the square root of -16 times the square root of -10?

May 19, 2018

$- 4 \sqrt{10}$

#### Explanation:

By definition of imaginary numbers: $i = \sqrt{- 1}$

$\sqrt{- 16} = 4 \cdot \sqrt{- 1} = 4 i$

$\sqrt{- 10} = \sqrt{10} \cdot \sqrt{- 1} = \sqrt{10} i$

${i}^{2} = i \cdot i = \sqrt{- 1} \cdot \sqrt{- 1} = {\left(\sqrt{- 1}\right)}^{2} = - 1$

so:

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