Adding or subtracting radicals is the same concept as that of adding or subtracting similar, or "like", terms. The index and the value under the radical (the radicand) must be the SAME (creating "like radicals") before you can add or subtract the radical expressions.

Adding and subtracting radicals: For radicals having the same index and the same values under the radical (the radicands), add (or subtract) the values in front of the radicals and keep the radical.

rad 1 add
ANSWER: rad1a
Since the radicals are the same, add the values in front of the radical symbols, and keep the radical. Do NOT add the values under the radicals. Think of having three of the radical 5s, adding 4 more of the radical 5s, and getting a total of 7 radical 5s.    rad AS1


ANSWER: rad 2 c
The radicals are different and each is already in simplest form. There is simply no way to combine these values. The answer is the same as the original problem.

rad add2
ANSWER: rad2b
At first glance, it appears that combining these terms under addition is not possible since the radicals are not the same. But if we look further, we can simplify the second term so it will be a "like" radical:
rad a 2

rad as4
ANSWER: rad as4 ans
  There is an implied "1" in front of rad3 as. All radicals are already in simplest form. Combine the "like" radicals. rad SA4 aa

rad a 4
Simplify the radicals first, and then subtract and add.

ANSWER: radas6ab
Notice that this problem mixes cube roots with a square root.

You cannot combine cube roots with square roots when adding.
They are not "like radicals".


Expressions with Variables
(Assume variables to be positive.)
ex1 as1
  Since the radicands are the same, add and subtract the coefficients (the numbers in front of the radicals).

ex2 as3 ANSWER: as5
  Simplify each term first. Then see if the expressions can be added.as4
Simplifying showed that we had similar radicals which could be added.

ex3 as7 ANSWER: as8
  Simplify first.
Each term simplified to show similar radicals which could be subtracted.


bullet REMEMBER: Always simplify first! When the radicals in an addition or subtraction problem are different, be sure to check to see if the radicals can be simplified. It may be the case that when the radicals are simplified, they will become "like" radicals, making it possible for them to be added or subtracted.



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