How to Simplify Radical Expressions With Addition

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Expressions involving radicals can look a bit daunting. In this lesson, we are going to look at how to simplify radical expressions using addition. We will look at some different examples and talk about the steps involved in this process.

Adding Radical Expressions

Do you love animals? I adore them! Suppose you and I are volunteering at a local animal shelter, and I have a group of cats and dogs from which I ask you to take 1 cat and then add another cat to that. In other words, I ask you to add 1 cat plus 1 cat, and tell me what you get. You would probably easily see that you have two cats.


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Now, suppose I asked you to add together 1 dog plus 1 dog. Same thing, it's easy to see that you would have 2 dogs.


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Okay, now what if I asked you to add together 1 cat and 1 dog.


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Hmmmm, you can't say you have 2 cats or 2 dogs or even 2 catdogs (there's no such thing!). You can only say that you have 1 cat and 1 dog. No matter how you look at it, cats and dogs are not alike, so you can only say you have x cats and y dogs in a group of cats and dogs. We see that to add two animals together, the animals have to be the same type of animal. The same rules apply to adding radical expressions.

Radical expressions are expressions that contain radicals. A radical of a number represents a number that when multiplied by itself the indicated number of times gives the number inside the radical. This, along with the various labeled parts of a radical, are illustrated in the following image.


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A radical can have any number as its index, but in this lesson, we are going to concentrate on square roots, but all the processes we use can be extended to other radicals.

We can add radical expressions the same way we can add numbers or terms; however, when adding radical expressions, the radical part in each term of the expression must be the same, where the radical part of an expression is the part of the expression that is the radical and the radicand. For example, the following image shows some like radical parts and some unlike radical parts.


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When we have like radical parts, we can add up how many of those radical parts we have and perform addition as shown in the following image:


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We see that we can simplify the expression down to one term by adding the like radical parts together. Okay, so when we are attempting to simplify a radical expression using addition, that is the first thing we want to look for - like radical parts.

Manipulating Radicals

So, what happens if we don't have like radical parts. I mean, as we saw, in the same way that we can't add cats and dogs, we can't add unlike radical parts. Don't worry, all hope is not lost. When we have an expression that we would like to simplify using addition, we may be able to manipulate the radicals to get like radical parts and then add as we did earlier. As we said, we're working with square root radicals, so the radical of a number is equal to the number that when multiplied by itself gives that number. For instance, sqrt(9)=3, because 3*3 = 9.

We can use this fact to manipulate radicals. To do this, we follow these steps:

  1. Factor what is under the radical completely.
  2. Pull any numbers that are listed twice in the factorization out in front of the radical. In general, pull any numbers that are listed the number of times that the index indicates out in front of the radical.

To see this more clearly, consider the square root of 8. We can manipulate this radical as follows:


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As another example, consider the square root of 32. Again, we can manipulate this radical as follows.


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We see that we can manipulate a radical expression to have a different radical part, but still have the same value.

Now comes the cool part! Notice that when we manipulated sqrt(8) and sqrt(32), we ended up with answers that have the same radical part? This is exactly what we would need to do if we wanted to use addition to simplify the expression sqrt(8) + sqrt(32)! We would manipulate the two terms to get the same radical part, and then we would add accordingly.


simpradadd8


We see we can simplify the radical expression sqrt(8) + sqrt(32) to 6sqrt(2). That's not so hard, is it?

Simplifying Radical Expressions Using Addition

Let's pull this process together in a nice series of steps. When we want to simplify a radical expression using addition, we take these steps.

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