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Adding and Subtracting Radical Expressions

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Hi, and welcome. My name is Anthony Varela. And today, we're going to add and subtract radical expressions. So first we're going to talk about like-terms and relate them to like-radicands. Then we're going to add and subtract with like-radicands. And then we'll add and subtract with unlike-radicands, and see what we have to do.

So, first, I'd like to talk about combining like-terms. This is something you might have heard before. This might be new to you. But, basically, what we mean by combining like-terms is if we have an expression like 3x plus 5x, well, 3x and 5x are what we call "like-terms" because we call them both "x-terms." They just have coefficients in front of x. And we can say then that 3x plus 5x equals 8x because 3 plus 5 equals 8.

So what does this have to do then with radical expressions? Well, take a look at this expression. 2 times root 3 plus 4 times root 3, here we have like-radicands. Taking a look at what's underneath the square root here, it's the same-- root 3 and root 3.

So we can add these two together by just adding 2 and 4 and then attaching our root 3. So this would be 6 times root 3. So with like-radicands, we can add or subtract them just like we combine like-terms.

So let's take a look at this expression here. We have negative 3 root 2 plus 5 root 3 plus 4 root 2. So what I want to do first is identify if I have any like-radicands. So I'm looking for the same thing underneath the radical. And I see two terms have like-radicands. So I can add or subtract them like I do combining like-terms.

And here, what I'm looking for then is the number outside of the radical negative 3 and positive 4. And so I'm going to add those two together. Negative 3 plus 4 is 1. So this is 1 times the square root of 2. And then I have to attach my plus 5 root 3.

So you notice then with unlike-radicands, we can't simply add or subtract them like we do combining like-terms. So what do we do then if we'd like to add or subtract with unlike-radicands? Here we have root 5 plus root 45. Those are not like-radicands.

So one strategy is to look for factors that you can break down a number and maybe you might reveal your own like-radicand. So I'm going to take 45 and I'm going to break it down. I know that 45 equals 9 5. Well, thanks for our product property of radicals, I can break this down into two square roots.

So I can say that the square root of 9 times 5 equals the square root of 9 times the square root of 5. And now I have a perfect square. The square root of 9 evaluates to an integer. So I'm going to write that down. Then I have root 5 plus 3 times root 5.

And now look what I have. I have like-radicands. So I can add these together. Here I have an implied 1. Here we have a 3. So 1 plus 3 is 4. So this combines to 4 times root 5. So look for factors to create your own like-radicands. That's one strategy when you're adding and subtracting with unlike-radicands.

All right. So now let's practice. Here we have 3 root 7 plus 4 root 7 minus 5 root 7. And so the first thing I notice is that I have like-radicands throughout my entire expression. They're all root 7, root 7, root 7. So really all I'm concerned about, then is combining these numbers that are outside of my radicals-- 3 plus 4 minus 5.

Well, 3 plus 4 is 7. Take away 5, that gives me 2. So I know that all of this then combines to 2 times root 7. That's probably a little bit too easy. So let's do a more complicated problem.

Here I have 2 times root 2 minus 3 times root 8. What do I do? Well, I want to break 8 down into some factors. So I know that 8 equals 4 times 2. So I'm going to replace 8 with 4 times 2 here. So I have 2 times root 2 minus 3 times the square root of 4 times 2.

Well, I know that due to our product property of radicals, I can split this up into two radicals. So the square root of 4 times 2 equals the square root of 4 times the square root of 2. So let me make that replacement.

I see a perfect square. The square root of 4 evaluates to an integer that equals 2. So let me make that replacement. Now I see integers 3 and 2 that I can multiply together. I can technically say negative 3 times 2.

So I'm going to be then subtracting a 6 root 2 here. And now I notice that I have like-radicands. I can combine these. 2 minus 6 equals negative 4. So 2 root 2 minus 6 root 2 equals negative 4 root 2.

So what did we talk about today with adding and subtracting radical expressions? You're going to encounter either like-radicands or unlike-radicands. With our like-radicands, we could add or subtract them just like we do when we combine like-terms.

So here's an example. If we have a root x plus b root x, because we have like-radicands, we can add a and b and multiply that by root x. With unlike-radicands, we cannot add or subtract like we do when we're combining like-terms. So we're going to look for factors to create our own like-radicands.

So thanks for watching this tutorial on adding and subtracting radical expressions. Hope to catch you next time.