# How do you simplify -3sqrt45+2sqrt12+3sqrt6-3sqrt20?

May 26, 2017

$= - 15 \sqrt{5} + \sqrt{3} \left(3 \sqrt{2} + 4\right)$

#### Explanation:

$- 3 \sqrt{45} + 2 \sqrt{12} + 3 \sqrt{6} - 3 \sqrt{20}$

Write the radicands (see below) as the product of their factors, using squares if possible:

$- 3 \sqrt{9 \times 5} + 2 \sqrt{4 \times 3} + 3 \sqrt{2 \times 3} - 3 \sqrt{4 \times 5}$

Find the roots where possible:

$= - 3 \cdot 3 \sqrt{5} + 2 \cdot 2 \sqrt{3} + 3 \sqrt{2} \sqrt{3} - 3 \cdot 2 \sqrt{5}$

Identify the like terms:

$= \textcolor{b l u e}{- 3 \cdot 3 \sqrt{5} - 3 \cdot 2 \sqrt{5} \text{ }} \textcolor{red}{+ 2 \cdot 2 \sqrt{3} + 3 \sqrt{2} \sqrt{3}}$

=$\textcolor{b l u e}{- 9 \sqrt{5} - 6 \sqrt{5}} \text{ } \textcolor{red}{+ 4 \sqrt{3} + 3 \sqrt{2} \sqrt{3}}$

$= - 15 \sqrt{5} + \sqrt{3} \left(3 \sqrt{2} + 4\right)$

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$\rightarrow$ the word for the value under a root sign is the 'radicand'

" "sqrt("radicand")