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Adding and Subtracting Radical Expressions

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Today we're going to talk about adding and subtracting radical expressions. A radical expression is just an expression, so either a number or variable or a combination of both. And that expression has a radical sign in it.

When we're adding and subtracting radical expressions, we need to think about the like terms. So we'll start by reviewing what it means to combine like terms. And then we'll do some example adding and subtracting radical expressions.

Let's start by reviewing combining like terms. Like terms have the same variable and exponent. And sometimes you will have a term that doesn't have an exponent, in which case you can combine it with something else without an exponent. And if it doesn't have a variable, then you can combine it with something else that also doesn't have a variable.

In this example I've got 3x squared. And I can combine that with the like term, 2x squared. They both have the same variable x and the same exponent too.

When I combine them, I'm just going to add them with the number in front. That's the only thing that's going to change about the terms. So 3x squared plus 2x squared will give me 5x squared.

Now I've got 5y and a minus 3y, which we can think of this as negative 3y. So they both have the same variable, and they both do not have an exponent. So that means we can combine them. So 5 and a negative 3 is going to give me a positive 2y.

And then lastly, we have just a regular number. It has no variable and no exponent. So as I said, we can combine that with something else with no variable and no exponent. So a positive 9 and a negative 2 will give me a positive 7.

Let's look at what it means to be a like term with a radical. Let's do some examples. When we are combining radicals by either adding or subtracting, we still need to think about our like terms. And a radical is a like term with another radical, if the radicand-- or the number underneath the radical-- is the same, and the index of the radical is also the same.

So here my radicands are both 3. And they are both square roots, so that means the index is the same. So these two radicals are like terms, so I can go ahead and combine them together by simply adding the numbers that are in front. So 5 square root of 3 plus 3 square root of 3 is going to give me 8 square root of 3.

Let's look at another example. Here I've got three terms that are being added or subtracted together, and I'm going to look for my like terms. I see here that they are all square roots, so they all have an index of two.

However, they do not all have the same radicand. So I can't combine them if they do not have the same radicand. They wouldn't be like terms.

I do see that these two have 8 as a radicand. So I can add the numbers together that are in front, so 3 square root of 8 plus 2 square root of 8 gives me 5 square root of 8. However, because the last term has a different radicand, I can't combine it. So I'm just going to bring that down as minus 4 square root of 5.

By simplifying this expression, I have 5 square root of 8 minus the 4 square root of 5. And that's as simplified as it can be.

For my third example I'm adding together two unlike radicals. Because these two radicals do not have the same radicand, we cannot simply just add them together. However, I do notice that 3 is a factor of 12. So I can rewrite the square root of 12 as the square root of 3, multiplied by the square root of another number.

Because 12 is equal to 3 times 4, using my product property of radicals I know that I can rewrite this as the square root of 3 times the square root of 4. And then, I notice that 4 is a perfect square. So that means that the square root of 4 is just going to give me 2. So this is just going to be equal to the square root of 3 times 2.

So now I'm going to bring down everything else and see what I've got. 6 square root of 3 plus 2 times the square root of 3 times another 2. So because multiplication is commutative, I know that I can multiply-- or I can evaluate this as 2 times 2 times the square root of 3. 2 times 2 just gives me 4, and then times the square root of 3.

And now that both of my radicals have the same radicand, I know that they are like terms, so I can add them together. 6 square root of 3 plus 4 square root of 3 will just give me 10 square root of 3.

Let's do a couple more examples. Here I've got negative 3 square root of 10 plus 6 square root of 10 minus 4 square root of 10. And because all of my radicals have the same radicand, and they also have the same index, they're all square roots. Then that means I can just combine them together.

So negative 3 plus 6 minus 4. Negative 3 plus 6 will give me 3 square root of 10. And 3 square root of 10 minus 4 square root of 10 will give me a negative 1 square root of 10.

Now I could also rewrite negative 1 square root of 10 as just the negative square root of 10. And this would be my final answer.

Let's look at a final example. I've got the square root of 20 minus 7 square root of 45 minus the square root of 5. I can see that none of my radicands are the same, so I might not be able to combine them. But I also notice that my smallest radicand is a factor of the other two radicands. So let's see if we can break those down and simplify, so that these first two radicals have a radicand of 5.

The square root of 20 could also be written as the square root of 4 times 5. And the square root of 45 could be written as the square root of 9 times 5. So using our product property of radicals, I can rewrite this as the square root of 4 times the square root of 5. I can rewrite this as the square root of 9 times the square root of 5. And I'll bring down my last term.

Now I notice that both 4 and 9 are perfect squares, which means that there square root is going to be an integer. The square root of 4 is 2. So this becomes 2 square root of 5. The square root of 9 is 3, so this becomes 3 square root of 5. And I'll bring down my last term.

So I had a number in front of this square root, so I'm going to want to simplify these two numbers by multiplying them together before I can combine these radicals. So 7 times 3 is going to give me 21. And now that I have simplified the multiplication, I can just subtract.

So 2 minus 21 is going to give me a negative 19 square root of 5. And negative 19 square root of 5 minus one more square root of 5 is going to give me negative 20 square root of 5 for my final answer.

Let's go over our key points from today. As usual, make sure you get these down in your notes if you don't have them already, so you can refer to them later. So we first talked about that when you're adding and subtracting radicals, you can combine them if they are like terms.

We saw that radicals are like terms if they have the same radicand and the same index. So they have to have the same number underneath the radical sign, and they have to have the same index, meaning they're both a square root or a cubed root or a fifth root. And then we saw that you can sometimes break down a radicand into factors to simplify the radicand, in which case you might be then able to combine it with another like term.

I hope that these key points and the examples that we did helped you understand a little bit more about adding and subtracting radicals. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.