Adding and subtracting radicals is very similar to adding and subtracting with variables. Consider the following example:
Notice that when we combined the terms with √11 it was just like combining terms with x. When adding and subtracting with radicals we can combine like radicals just as like terms. We add and subtract the coefficients in front of the radical, and the radical stays the same. This is shown in the following example.
We cannot simplify this expression any more as the radicals do not match. Often problems we solved have no like radicals, however, if we simplify the radicals first we may find we do in fact have like radicals.
The Arab writers of the 16th century used the symbol similar to the greater than symbol with a dot underneath for radicals.
This exact process can be used to add and subtract radicals with higher indices. ("Indices" is the plural of index, which indicates the type of root).
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html