Welcome back, everyone. So as we've talked about square roots and cube roots, we've seen lots of perfect powers, like perfect squares or cubes. So, for example, like the square root of 9, which is 3, or the cube root of 8, which is 2. But in a lot of problems, that's not going to happen. In a lot of problems, you might see something like the square root of 20 or the cube root of 54, and you're going to have to know how to take those expressions and make them simpler. That's what I'm going to show you how to do in this video, and it turns out that one of the ways that we can simplify radicals is actually by expanding them, sort of like making them a little bit bigger so that we can hopefully make them smaller and simpler later on. I'm going to show you how to do this. It's very straightforward. Let's get started. So, basically, when radicals aren't perfect powers like the radical twenty, the whole thing we're going to do is we're going to simplify it by turning it into a product. We're going to try to break it up into a product of 2 things, and the whole goal is that one of the terms will be a perfect power. So here's the thing. I'm going to take the radical 20, and I want to break it up so that it's the product of 2 things, and I want one of these things to be a perfect power, like 4 or 9 or 16 or something like that. Alright? So I have this table here. We're going to just go down this table and see if we can turn the 20 into a product where 4 or 9 is one of the terms. So can this happen? So can I do 4? Well, if you take a look at 4 times 5, that equals 20. So, basically, you just separated this thing into 2 radicals, and I can totally do that. So what's the square root of 4? We have already seen that it's just 2. What's the square root of 5? Well, that's just the square root of 5, and that's not a perfect number. So, basically, what we've seen here is that we've turned this into a 2 times radical 5, and so the simplest way that we can rewrite this expression is just 2√5. Now can we go any further? No. Because 5 is just a prime number, so we can't break that radical up any further. So we say this expression is fully simplified because we can't break up the radicals any further than we already have. Alright? But that's the basic idea. So, as a formula, the way that you're going to see this in your textbook is if a number in a radical has factors a and b, you basically can just break it up into a and b, and then you can split them up into their own radicals, like radical of a and then radical of b. And then you can just deal with those separately. Alright? That's the whole thing. Let's move on to the second problem now because in some problems, you're going to have variables as well. So what I like to do is I like to separate this thing into the number times the variable. Alright? So, again, when I take these radicals, can I break them up into anything in which we're going to get a perfect square out of it? Well, let's do the 18 first. So could 18 does 18 reduce to anything? So does 4 go into 18? Well, 4 times 4 is 16. 4 times 5 is 20, so it doesn't. What about 9? Well, actually, 18 could be written as the product of radical 9 times radical 2. Right? That separates. And then what about the x squared? Well, I have the square roots of x squared. So it turns out that, actually, the square root of x squared is a perfect power, or sorry, the x squared is a perfect power. And, basically, what happens is you've just undone the exponents. So this actually just turns into an x over here. Alright? Now are we done yet? Is this our full expression? Well, no, because the square root of 9 actually just turns into a 3. So what about the radical 2? Can we break up the radical 2 any further? No, because it's just a prime number. And so, basically, what happens is this is our simplest that we can write this expression, and what you're going to see here is that the x usually gets moved in front of the radical. So this whole thing really just becomes 3x√2 and that is our fully simplified expression.
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Simplifying Radical Expressions
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