# How can you simplify #sqrt(28-5sqrt(12))# ?

##### 1 Answer

#### Explanation:

For a start:

#sqrt(12) = sqrt(2^2*3) = 2sqrt(3)#

So:

#sqrt(28-5sqrt(12)) = sqrt(28-10sqrt(3))#

Can this be simplified further?

Let us attempt to find rational

#28 - 10sqrt(3) = (a+bsqrt(3))^2#

#color(white)(28 - 10sqrt(3)) = (a^2+3b^2)+2a b sqrt(3)#

Equating coefficients:

#{ (a^2+3b^2 = 28), (2ab = -10) :}#

From the second equation, we find:

#b = -5/a#

Substituting

#28 = a^2+75/a^2#

Subtracting

#0 = (a^2)^2-28(a^2)+75#

#color(white)(0) = (a^2-25)(a^2-3) = (a-5)(a+5)(a-sqrt(3))(a+sqrt(3))#

Since we want

#a = +-5#

If

If

Since we want the non-negative square root, we want