# How do you simplify (sqrt3-sqrt2)(sqrt15+sqrt12)?

May 29, 2017

See a solution process below:

#### Explanation:

To multiply these two terms you multiply each individual term in the left parenthesis by each individual term in the right parenthesis.

$\left(\textcolor{red}{\sqrt{3}} - \textcolor{red}{\sqrt{2}}\right) \left(\textcolor{b l u e}{\sqrt{15}} + \textcolor{b l u e}{\sqrt{12}}\right)$ becomes:

$\left(\textcolor{red}{\sqrt{3}} \times \textcolor{b l u e}{\sqrt{15}}\right) + \left(\textcolor{red}{\sqrt{3}} \times \textcolor{b l u e}{\sqrt{12}}\right) - \left(\textcolor{red}{\sqrt{2}} \times \textcolor{b l u e}{\sqrt{15}}\right) - \left(\textcolor{red}{\sqrt{2}} \times \textcolor{b l u e}{\sqrt{12}}\right)$

Next, use this rule of radicals to multiply the four sets of radicals:

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

$\sqrt{3 \times 15} + \sqrt{3 \times 12} - \sqrt{2 \times 15} - \sqrt{2 \times 12}$

$\sqrt{45} + \sqrt{36} - \sqrt{30} - \sqrt{24}$

$\sqrt{45} + 6 - \sqrt{30} - \sqrt{24}$

Now, we can rewrite the expression to simplify as:

$\sqrt{9 \times 5} + 6 - \sqrt{30} - \sqrt{4 \times 6}$

$\left(\sqrt{9} \times \sqrt{5}\right) + 6 - \sqrt{30} - \left(\sqrt{4} \times \sqrt{6}\right)$

$3 \sqrt{5} + 6 - \sqrt{30} - 2 \sqrt{6}$