Hey, everyone. We've talked a lot about radicals so far. One of the things that we haven't seen yet is how to add and subtract expressions that have radicals. That's what we're going to go over in this video. It turns out that it's actually very similar to how we dealt with algebraic expressions, and I'm going to show you it's very straightforward. So let's check it out here.
So when we dealt with algebraic expressions, if I had something like x2+3 plus 8, the way I simplified this was I combined like terms. I could combine the 2xs and the 4xs and the 3s and the 8s, and this just became 6x plus 11. The idea is that when I have a radical expression, instead of combining like terms, I'm gonna combine like radicals. So like radicals just means that they have the same radicand. They have the same thing inside of the radicals, in other words, x and x in this problem, and they also have the same index. So I have to make sure that they're both square roots and not that one is a square root and one is a cube root or something like that. As long as we have the same radicand and the same index, we can just add them. And how do we add them? We add them exactly how we added 2x and 4x. So 2 times the square root of x and 4 times the square root of x. It's kind of like I'm adding apples and apples. Right? So this just becomes 6 times the square root of x, so that's 6√x, and then the 3 and 8 just become 11 how they like, just like they always have. So this is how you simplify these kinds of expressions. You can only combine things that are like each other. Alright? That's all there is to it. So let's go ahead and take a look at a couple more examples here.
Alright. So if we have, let's say, 3 √7 times 2 plus 2 square root of 7 minus the cube root of 7, how do I simplify and add this expression? Well, remember, I can only combine the like radicals. Like radicals have like radicands and the same index. They have the same thing under the radical and the same index. Here, what we have is we'll notice that all of the radicands are 7s. I have sevens in all of the symbols over here. But are all the indexes the same? Well, no. Because here I have an index of 3, whereas here I have square roots. And remember, those are indexes of 2. So these two things have the same radicand and the same index, but this one has the same radicand but a different index. So it's not a like term. Okay? So just be very careful when you're doing that. So basically, what happens is I can combine the 2 things in yellow. So the 3 and the 2 just combine down to 5 radical 7 or square root of 7, and then I have over here minus the cube root of 7. So this is like an apple, and the cube root of 7 is like a banana. I can't add those things because they're not the same. And so this is how I simplify my expression, and that's the answer. Alright? So pretty straightforward. Now what I want to warn you against, actually, is something that I see a lot of students get this you know, make this mistake. Basically, whenever you see radicals that are, you know, separate from each other and you're adding them, you can't combine them into one radical. So, for example, I can't take the square root of 7 and the square root of 7. That's not equal to the square root of 14. This is a mistake I see a lot of students make. Just be very careful that you don't do this. Otherwise, you're gonna get the wrong answer. Radical 7+7 does not equal radical 14. You can't just merge that stuff into the same radical. Alright? Just be very careful.
Alright. So let's look at another example here. Here, we have 9 times the cube root of x, and then we have a square root of x. So, here, we have to combine the like terms. So, if you notice here, I've got the same radicands. I've got xs everywhere, but I have different indexes. Here, what I have is I have a cube root of x and a cube root of x, and then here I have a square root of x, so those are different. They're not like radicals. Then and I just have a constant over here. So what can I combine? I can combine the 9 and the 4, 9 minus 4. Remember, just keep the sign over here. This becomes 4 times the cube root of x. Then I have minus the square root of x, then I have plus 3. So that's how to simplify this expression. Alright? So that's all there is to it, folks. Let me know if you have any questions, and thanks for watching.