How can you determine wheter the decimal form of a radical represents its exact value?

It's pretty hard to do that. If you know what radical base it is (say I know I'm going to have a √3 in my answer) then you can divide the decimal (let's say I have 0.866025404 ...) by the base number in order to get the constant in front of the radical (0.866025404 / √3 ≈ 0.5 = 1/2 → √ (3) / 2).