# How do you simplify 4sqrt112+ 5sqrt56- 9sqrt126?

Jun 30, 2018

See a solution process below:

#### Explanation:

First, rewrite each of the radicals as:

$4 \sqrt{16 \cdot 7} + 5 \sqrt{4 \cdot 14} - 9 \sqrt{9 \cdot 14}$

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

$4 \sqrt{16} \sqrt{7} + 5 \sqrt{4} \sqrt{14} - 9 \sqrt{9} \sqrt{14} \implies$

$\left(4 \cdot 4\right) \sqrt{7} + \left(5 \cdot 2\right) \sqrt{14} - \left(9 \cdot 3\right) \sqrt{14} \implies$

$16 \sqrt{7} + 10 \sqrt{14} - 27 \sqrt{14}$

Next, we can factor our common terms:

$16 \sqrt{7} + \left(10 - 27\right) \sqrt{14} \implies$

$16 \sqrt{7} + \left(- 17\right) \sqrt{14} \implies$

$16 \sqrt{7} - 17 \sqrt{14}$

If necessary we can go this additional step:

$16 \sqrt{7} - 17 \sqrt{2 \cdot 7} \implies$

$16 \sqrt{7} - 17 \sqrt{2} \sqrt{7} \implies$

$\left(16 - 17 \sqrt{2}\right) \sqrt{7}$