Question

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

Answer 1

`sqrt(28-5sqrt(12)) = 5 - sqrt(3)`

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 `a, b` such that:

`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 `-5/a` for `b` in the first equation, we get:

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

Subtracting `28` from both sides and multiplying by `a^2` we get:

`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` to be rational, we have:

`a = +-5`

If `a = 5` then `b = -5/a = -1`, resulting in `5 - sqrt(3)` which is positive.

If `a = -5` then `b = -5/a = 1`, resulting in `-5 + sqrt(3)` which is negative.

Since we want the non-negative square root, we want `5 - sqrt(3)`