# How do you simplify 3sqrt20 + 5sqrt45 + sqrt75?

Mar 31, 2017

$3 \sqrt{20} + 5 \sqrt{45} + \sqrt{75} = 21 \sqrt{5} + 5 \sqrt{3}$

#### Explanation:

A square root can be simplified by looking for a square number that's a factor of the number within a square root. We use the rule $\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}$ to simplify square roots.

Looking at just the first term, we see that we have $\sqrt{20}$. This can be simplified by noticing that $20 = 4 \times 5$, and $4 = {2}^{2}$. Then, we can say that $\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \sqrt{5} = 2 \sqrt{5}$.

We can follow this logic for all the square roots:

$3 \sqrt{20} + 5 \sqrt{45} + \sqrt{75}$

$= 3 \sqrt{4 \times 5} + 5 \sqrt{9 \times 5} + \sqrt{25 \times 3}$

$= 3 \sqrt{4} \sqrt{5} + 5 \sqrt{9} \sqrt{5} + \sqrt{25} \sqrt{3}$

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

$= 6 \sqrt{5} + 15 \sqrt{5} + 5 \sqrt{3}$

The terms that both contain $\sqrt{5}$ can be simplified. This is analogous to doing $6 x + 15 x = 21 x$:

$= 21 \sqrt{5} + 5 \sqrt{3}$

Mar 31, 2017

$21 \sqrt{5} + 5 \sqrt{3}$
$3 \sqrt{20} = 3 \sqrt{4 \cdot 5} = 3 \cdot 2 \sqrt{5} = 6 \sqrt{5}$
$5 \sqrt{45} = 5 \sqrt{9 \cdot 5} = 5 \cdot 3 \sqrt{5} = 15 \sqrt{5}$
$\sqrt{75} = \sqrt{25 \cdot 3} = 5 \sqrt{3}$