# How do you simplify sqrt(-50)?

Jul 5, 2015

I found: $\sqrt{- 50} = 5 i \sqrt{2}$

#### Explanation:

Here you have a problem...
in fact you cannot solve a square root with a negative argument or, better, you cannot find a real number which is solution of your square root.

What you can do it is try to find a solution somewhere else...in the place of imaginary numbers!
First of all you manipulate your root as:
$\sqrt{- 50} = \sqrt{- 1 \cdot 2 \cdot 25} = \sqrt{- 1} \sqrt{25} \sqrt{2} = 5 \sqrt{- 1} \sqrt{2}$
you now introduce a new entity: the imaginary unit $i$ given as:
$i = \sqrt{- 1}$
so you can write:
$\sqrt{- 50} = 5 i \sqrt{2}$ which is:

1] solution of your problem; in fact if you square it you get:
${\left(5 i \sqrt{2}\right)}^{2} = 25 \cdot - 1 \cdot 2 = - 50$ that is your original radicand!

2] it is an imaginary number (it contains $i$).