# What is (square root of [6] + 2 square root of [2])(4square root of [6] - 3 square root of 2)?

Sep 17, 2015

$12 + 5 \sqrt{12}$

#### Explanation:

We multiply cross-multiply, that is,
$\left(\sqrt{6} + 2 \sqrt{2}\right) \left(4 \sqrt{6} - 3 \sqrt{2}\right)$
equals
$\sqrt{6} \cdot 4 \sqrt{6} + 2 \sqrt{2} \cdot 4 \sqrt{6} - \sqrt{6} \cdot 3 \sqrt{2} - 2 \sqrt{2} \cdot 3 \sqrt{2}$

The square roots time themselves equal the number under the root, so
$4 \cdot 6 + 8 \sqrt{2} \sqrt{6} - 3 \sqrt{6} \sqrt{2} - 6 \cdot 2$

We put $\sqrt{2} \sqrt{6}$ in evidence:
$24 + \left(8 - 3\right) \sqrt{6} \sqrt{2} - 12$

We can join these two roots in one, after all $\sqrt{x} \sqrt{y} = \sqrt{x y}$ as long as they're not both negative. So, we get
$24 + 5 \sqrt{12} - 12$

Finally, we just take the difference of the two constants and call it a day
$12 + 5 \sqrt{12}$