# How do you simplify 12sqrt15 - 9sqrt75 + 8sqrt45 - 6sqrt225?

Jul 9, 2017

See a solution process below:

#### Explanation:

First, we can rewrite the expression as:

$12 \sqrt{15} - 9 \sqrt{25 \cdot 3} + 8 \sqrt{9 \cdot 5} - \left(6 \cdot 15\right) \implies$

$12 \sqrt{15} - 9 \sqrt{25 \cdot 3} + 8 \sqrt{9 \cdot 5} - 90$

Now we can rewrite the radicals using this rule for radicals:

$\sqrt{\textcolor{red}{a} \cdot \textcolor{b l u e}{b}} = \sqrt{\textcolor{red}{a}} \cdot \sqrt{\textcolor{b l u e}{b}}$

$12 \sqrt{15} - 9 \sqrt{\textcolor{red}{25} \cdot \textcolor{b l u e}{3}} + 8 \sqrt{\textcolor{red}{9} \cdot \textcolor{b l u e}{5}} - 90 \implies$

$12 \sqrt{15} - 9 \sqrt{\textcolor{red}{25}} \sqrt{\textcolor{b l u e}{3}} + 8 \sqrt{\textcolor{red}{9}} \sqrt{\textcolor{b l u e}{5}} - 90 \implies$

$12 \sqrt{15} - \left(9 \cdot 5\right) \sqrt{\textcolor{b l u e}{3}} + \left(8 \cdot 3\right) \sqrt{\textcolor{b l u e}{5}} - 90 \implies$

$12 \sqrt{15} - 45 \sqrt{\textcolor{b l u e}{3}} + 24 \sqrt{\textcolor{b l u e}{5}} - 90 \implies$