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.
Add/Subtract
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.
ANSWER:
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.
ANSWER:
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.
ANSWER:
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:
ANSWER:
There is an implied "1" in front of
. All radicals are already in simplest form. Combine the "like" radicals.
ANSWER:
Simplify the radicals first, and then subtract and add.
ANSWER:
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.)
ANSWER:
Since the radicands are the same, add and subtract the coefficients (the numbers in front of the radicals).
ANSWER:
Simplify each term first. Then see if the expressions can be added.
Simplifying showed that we had similar radicals which could be added.
ANSWER:
Simplify first.
Each term simplified to show similar radicals which could be subtracted.
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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