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

##### 1 Answer

Oct 27, 2015

Find rational solution of

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

#### Explanation:

Is

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

So are there rational numbers

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

From

Substitute that into

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

Subtract

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

This has rational roots

We might as well choose

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

So:

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