Radicals are nothing but square roots. Radicals can of course be added, but only with like numerals. This means the number inside the square must be the same in all cases. A fraction can contain radical, but it is not a radical as such. The confusion between radical and fraction arises because of the word rational. What is the meaning of rational? It is nothing but a fraction without a radical. There is absolutely no relation between rational and radical and hence beginners be careful when you deal with it!
When the values inside the root are the same it can be reduced easily. For instance, consider the following equation
5 Root 8x + 4 Root 8x
As said the values inside the square are the same and hence you can just add it like any other number which means the answer is
9 Root 8x
It can further be simplified if required. Simplification means the root has to be split further to its basic form. Here it would be like this
8x is equal to Root 4x * Root 2x
The square root of 4 is 2*2, as it is a perfect square. So the equation becomes
9*2 Root 2
That means 18 Root 2
If not alike you can try to simplify as far as possible and then add. For instance, consider the following equation:
5 Root (40x) + 3 Root (10x)
The above is an equation with dissimilar roots that means it cannot be added. But you can do the same by making it simpler. Consider the first part of the equation alone which is 5 Root 40. Here 40 can be simplified as 10*4 that means Root10 multiplied by Root 4. 4 is a perfect square and hence you can obviously abridge it to 2.
5 Root (10x*4x)
The equation is now as listed below
5*2 Root (10x) +3 root (10x)
Further 5 and 2 can be multiplied that gives 10 as the result. Now the equation is reduced to 10 Root 10 and similarly the other part is 3 Root 10. You can now add them effortlessly as they are alike!
10 Root (10x) +3 Root (10x)
13Root (10)
There can only be two categories when trying to add radicals and the basics are for sure explained here clearly. The like ones are not at all an issue. Just go a head and add them like any other number. When they are dissimilar, you have to reduce and then proceed with addition.

Tech-FAQ is reliable source for information related to How to Add Radicals and how to add square roots.