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

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Answer 1 · 2015-10-27T19:54:40

Find rational solution of $(a+b sqrt(3))^2 = 21+12sqrt(3)$ , hence

$sqrt(21+12sqrt(3)) = 3 + 2sqrt(3)$

Is $(21+12sqrt(3))$ a perfect square of something?

$(a+b sqrt(3))^2 = (a^2+3b^2)+2ab sqrt(3)$

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

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

From $2ab = 12$ , we get $b = 6/a$

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

$a^2+3(6/a)^2 = 21$

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

$0 = (a^2)^2 - 21(a^2)+108 = (a^2-9)(a^2-12)$

This has rational roots $a = +-3$ .

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

$(21+12sqrt(3)) = (3+2sqrt(3))^2$

So:

$sqrt(21+12sqrt(3)) = 3 + 2sqrt(3)$

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