To radicalise a square root you break the number down to its factors and then decide whether you can pair any of the factors. For example, if the number is 64, this can be broken down to 2*2*2*2*2*2, that is 3 pairs of twos. When you take the square root, each pair reduces to one single number: so we have (2*2)(2*2)(2*2) reducing to 2*2*2. This is the number 8.
But it's not always that easy. Another example, 180=2*2*3*3*5=(2*2)(3*3)5. This time we can reduce the pairs to 2*3=6, but we have a 5 left. To radicalise the square root we take the reduced pairs, 6, and multiply by the square root of 5: 6sqrt(5). That is square root of 180 in radical form.
Now let's take 60=2*2*3*5=(2*2)3*5. Square root is 2sqrt(15). Example: 1350=2*3*3*3*5*5=2(3*3)3(5*5). We can reduce the pairs to 3*5 but we're left with 2*3=6; so the square root in radical form is 3*5sqrt(6)=15sqrt(6).
We can't radicalise negative numbers unless we involve the imaginary number i, defined as the square root of -1. But if we wanted to radicalise square root of -60, we would radicalise 60 then follow it with i: 2sqrt(15)i or 2isqrt(15).
Not all numbers can be radicalised, but that just means we can't pair any of its factors.
Other roots, like cube roots, can be radicalised, too, but instead of pairs we group in threes, so cube root of 1350=2(3*3*3)5*5. Reduce the group of three 3s to just one: 3cuberoot(50). But 64=(2*2*2)(2*2*2), so the cube root of 64 is 2*2=4 because each of three 2s reduces to 2. For fourth roots we group in fours, and so on.
Radicalisation happens in algebra, too. 25(x+1)(x-2)^2= (5*5)((x-2)*(x-2))(x+1), so square root is 5(x-2)sqrt(x+1).
