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There are specific rules governing adding and subtracting radical expressions. This lesson will describe these rules and give examples of how they are used.

One of These Things Is Not Like the Other

Remember that song from the kids show, it went something like, 'One of these things is not like the other. One of these things just doesn't belong?' Then, the point was to look at four items and determine which one of them was different, either because of color, shape or some other characteristic. There are times in your study of algebra when you will have to look at different terms and determine which of them are alike and which are different.

Like Terms with Radicals

Terms containing radicals, or square roots, are considered like terms if the portion of the terms under the radical symbols is the same. These are like terms:

These are not like terms:

Add and Subtract Radicals

In order to add and subtract expressions with radicals, they must be like terms. If the portions under each radical are different, they cannot be combined using addition or subtraction. If the portions under the radicals are the same, then to add or subtract simply add or subtract the number in front of the radical symbol leaving what is under the radical the same.

Here's an example. Simplify:

Because both of these terms have a square root of 2, they can be added together to get 6âˆš2. How about this one?

Again, since both terms have a 7 under the square root symbol, they can be combined to get 4âˆš7. Let's try one more example.

In this example, because one of the terms under the radical is a 5 and the other is a 2, they cannot be combined and the expression is as simplified as it can be.

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You can also simplify longer expressions that might have some terms that will combine and others that won't. Consider this example:

Some of these terms can be combined while others cannot. Sometimes it's easier to put different shapes around the like terms to set them apart. For example, you could circle the terms with the âˆš5, put a triangle around the terms with a âˆš3 and a square around the terms with a âˆš2. Then it's easy to see which terms need to be combined. When you combine the like terms, you get:

No more simplifying can be done.

Lesson Summary

When adding or subtracting radical expressions, it's important that the terms under the radicals are the same. If they are different, the terms cannot be combined. When they are like each other and can be combined, this is done by adding or subtracting the numbers in front of the radical. The term under the radical does not change.

Learning Outcomes

Watching the video lesson could prepare you to achieve the following goals:

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