# How do you simplify sqrt(-5) + sqrt(-20)?

Oct 16, 2015

$3 \sqrt{- 5} \text{ }$, or $\text{ } 3 i \sqrt{5}$

#### Explanation:

The trick here is to try and rewrite $- 20$ as

$- 20 = - 5 \cdot 4$

This means that $\sqrt{- 20}$ can be written as

$\sqrt{- 20} = \sqrt{- 5 \cdot 4} = \sqrt{- 5} \cdot \sqrt{4} = 2 \cdot \sqrt{- 5}$

The expression becomes

$\sqrt{- 5} + 2 \sqrt{- 5}$

These radical terms can be combined to give

$\sqrt{- 5} + 2 \sqrt{- 5} = \sqrt{- 5} \cdot \left(1 + 2\right) = 3 \sqrt{- 5}$

If you're familiar with complex numbers, you can further simplify this expression by using

${i}^{2} = - 1 \implies \sqrt{- 1} = i$

This will get you

$3 \sqrt{- 5} = 3 \sqrt{- 1 \cdot 5} = 3 \sqrt{- 1} \cdot \sqrt{5} = 3 \cdot i \cdot \sqrt{5}$