# How do you simplify sqrt(21+12sqrt3)?

Oct 27, 2015

Find rational solution of ${\left(a + b \sqrt{3}\right)}^{2} = 21 + 12 \sqrt{3}$, hence

$\sqrt{21 + 12 \sqrt{3}} = 3 + 2 \sqrt{3}$

#### Explanation:

Is $\left(21 + 12 \sqrt{3}\right)$ a perfect square of something?

${\left(a + b \sqrt{3}\right)}^{2} = \left({a}^{2} + 3 {b}^{2}\right) + 2 a b \sqrt{3}$

So are there rational numbers $a$ and $b$ such that:

${a}^{2} + 3 {b}^{2} = 21$ and $2 a b = 12$ ?

From $2 a b = 12$, we get $b = \frac{6}{a}$

Substitute that into ${a}^{2} + 3 {b}^{2} = 21$ to find:

${a}^{2} + 3 {\left(\frac{6}{a}\right)}^{2} = 21$

Subtract $21$ from both sides and multiply through by ${a}^{2}$ to get:

$0 = {\left({a}^{2}\right)}^{2} - 21 \left({a}^{2}\right) + 108 = \left({a}^{2} - 9\right) \left({a}^{2} - 12\right)$

This has rational roots $a = \pm 3$.

We might as well choose $a = 3$ hence $b = 2$ and we have found that:

$\left(21 + 12 \sqrt{3}\right) = {\left(3 + 2 \sqrt{3}\right)}^{2}$

So:

$\sqrt{21 + 12 \sqrt{3}} = 3 + 2 \sqrt{3}$