# How do you simplify sqrt(37-4sqrt(75)) ?

Jan 16, 2018

$\sqrt{37 - 4 \sqrt{75}} = 5 - 2 \sqrt{3}$

#### Explanation:

I think you mean the positive square root of $37 - 4 \sqrt{75}$

Note that:

$37 - 4 \sqrt{75} = 37 - 4 \sqrt{{5}^{2} \cdot 3}$

$\textcolor{w h i t e}{37 - 4 \sqrt{75}} = 37 - 20 \sqrt{3}$

$\textcolor{w h i t e}{37 - 4 \sqrt{75}} = {5}^{2} - 2 \left(5\right) \left(2 \sqrt{3}\right) + 3 \cdot {2}^{2}$

$\textcolor{w h i t e}{37 - 4 \sqrt{75}} = {\left(5 - 2 \sqrt{3}\right)}^{2}$

Also note that $2 \sqrt{3} \approx 3.464 < 5$, So $5 - 2 \sqrt{3} > 0$.

So:

$\sqrt{37 - 4 \sqrt{75}} = 5 - 2 \sqrt{3}$