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 as sort of, like, making them a little bit bigger so that we can hopefully make it 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 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 44 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'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 radical 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 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 does 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 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 our 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 radical 2, and that is our fully simplified expression. Now for the last one over here, we have the cube root of 54x to the 4th power. So now we no longer have square roots, we have cube roots, but the idea is the same. And, again, what I like to do here is break this up into 2 radicals. So radical 54, cube root 54, and then the cube root of x to the 4th power. And I'll just deal with those independently. Right? So let's do the 54 first. Can I break this up into a perfect cube? Perfect cubes are going to be over here, like 827 and stuff like that. So what about eights? Well, 8 doesn't go into 54. 8 times 7 is 56, so it's close. What about 27? 27 is actually, yeah, this does work. So in other words, this is just 27 ⨉ 2. So I've gotten a perfect cube out of this. Alright? Now what about the cube root of x to the 4th power? Is there a perfect cube that I can pull out of that? Well, think about it. This is just x multiplied by itself 4 times, so what I can do is I can just split this up into x cubed times just cube root of x. And the reason this is helpful is because if I have a cube root of a cube, then I just basically undo it. Right? So with this 27, the cube root of this turns turns out to be is just 3. Now the cube root of 2 doesn't simplify, but what about the cube root of x to the third power? This actually just becomes x, just like the square root of x to the second power became x. And then finally, the cube root of x over here is just left alone. Alright. So how do I make this simpler? Well, I just mash these 3 and the x together and just becomes 3x. And And one of the things you'll you'll see is that when you have these two expressions, you basically just put them back together again on the under the same radical. This just becomes the, cube roots of, this just becomes 2x over here. And this is the fully simplified expression. I can't break this up any further. Alright? So that's all there is to it, folks. Let me know if you have any questions. Let's get some practice.
Table of contents
- 0. Fundamental Concepts of Algebra3h 29m
- 1. Equations and Inequalities3h 27m
- 2. Graphs1h 43m
- 3. Functions & Graphs2h 17m
- 4. Polynomial Functions1h 54m
- 5. Rational Functions1h 23m
- 6. Exponential and Logarithmic Functions2h 28m
- 7. Measuring Angles39m
- 8. Trigonometric Functions on Right Triangles2h 5m
- 9. Unit Circle1h 19m
- 10. Graphing Trigonometric Functions1h 19m
- 11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m
- 12. Trigonometric Identities 2h 34m
- 13. Non-Right Triangles1h 38m
- 14. Vectors2h 25m
- 15. Polar Equations2h 5m
- 16. Parametric Equations1h 6m
- 17. Graphing Complex Numbers1h 7m
- 18. Systems of Equations and Matrices1h 6m
- 19. Conic Sections2h 36m
- 20. Sequences, Series & Induction1h 15m
- 21. Combinatorics and Probability1h 45m
- 22. Limits & Continuity1h 49m
- 23. Intro to Derivatives & Area Under the Curve2h 9m
0. Fundamental Concepts of Algebra
Simplifying Radical Expressions
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