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

statement
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.


expin1
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


expin2

rad3
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.


expin3
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


expin4
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


expin5
rad a 4
ANSWER: RAD4B
 
Simplify the radicals first, and then subtract and add.
rad4a


expin6
radas6
ANSWER: radas6ab
 
Notice that this problem mixes cube roots with a square root.
radas6a

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

 

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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