How do you perform the operation and write the result in standard form given sqrt(-5)*sqrt(-10)?

Feb 19, 2017

You could write $\sqrt{- 5} = \sqrt{\left(- 1\right) \cdot 5} = \sqrt{- 1} \cdot \sqrt{5}$ and do the same to the other radical.

Explanation:

$= \left(\sqrt{- 1} \cdot \sqrt{5}\right) \times \left(\sqrt{- 1} \cdot \sqrt{10}\right)$

$= \sqrt{- 1} \cdot \sqrt{- 1} \cdot \sqrt{5} \cdot \sqrt{10}$

$= {\left(\sqrt{- 1}\right)}^{2} \cdot \sqrt{5 \cdot 10} = - 1 \cdot \sqrt{50}$

And since $50 = 2 \cdot {5}^{2}$:

$= - 1 \cdot \sqrt{2} \cdot {\left(\sqrt{5}\right)}^{2} = - 5 \sqrt{2}$

But:
Something can be said for the following:

Since $\sqrt{A} \cdot \sqrt{B} = \sqrt{A \cdot B}$ we could do:

$= \sqrt{\left(- 5\right) \cdot \left(- 10\right)} = \sqrt{50} = + 5 \sqrt{2}$