Video transcript
What I want to do in this video is resimplify this expression, 3 times the principal root of 500 times x to the third, and take into consideration some of the comments that we got out on YouTube that actually give some interesting perspective on how you could simplify this. So just as a quick review of what we did in the last video, we said that this is the same thing as 3 times the principal root of 500. And I'm going to do it a little bit different than I did in the last video, just to make it interesting. This is 3 times the principal root of 500 times the principal root of x to the third. And 500-- we can rewrite it, because 500 is not a perfect square. We can rewrite 500 as 100 times 5. Or even better, we could rewrite that as 10 squared times 5. 10 squared is the same thing as 100. So we can rewrite this first part over here as 3 times the principal root of 10 squared times 5 times the principal root of x squared times x. That's the same thing as x to the third. Now, the one thing I'm going to do here-- actually, I won't talk about it just yet, of how we're going to do it differently than we did it in the last video. This radical right here can be rewritten as-- so this is going to be 3 times the square root, or the principal root, I should say, of 10 squared times the square root of 5. If we take the square root of the product of two things, it's the same thing as taking the square root of each of them and then taking the product. And so then this over here is going to be times the square root of, or the principal root of, x squared times the principal root of x. And the principal root of 10 squared is 10. And then what I said in the last video is that the principal root of x squared is going to be the absolute value of x, just in case x itself is a negative number. And so then if you simplify all of this, you get 3 times 10, which is 30-- and I'm just going to switch the order here-- times the absolute value of x. And then you have the square root of 5, or the principal root of 5, times the principal root of x. And this is just going to be equal to the principal root of 5x. Taking the square root of something and multiplying that times the square root of something else is the same thing as just taking the square root of 5x. So all of this simplified down to 30 times the absolute value of x times the principal root of 5x. And this is what we got in the last video. And the interesting thing here is, if we assume we're only dealing with real numbers, the domain of x right over here, the x's that will make this expression defined in the real numbers-- then x has to be greater than or equal to 0. So maybe I could write it this way. The domain here is that x is any real number greater than or equal to 0. And the reason why I say that is, if you put a negative number in here and you cube it, you're going to get another negative number. And then at least in the real numbers, you won't get an actual value. You'll get a square root of a negative number here. So if you make this-- if you assume this right here, we're dealing with the real numbers. We're not dealing with any complex numbers. When you open it up to complex numbers, then you can expand the domain more broadly. But if you're dealing with real numbers, you can say that x is going to be greater than or equal to 0. And then the absolute value of x is just going to be x, because it's not going to be a negative number. And so if we're assuming that the domain of x is-- or if this expression is going to be evaluatable, or it's going to have a positive number, then this can be written as 30x times the square root of 5x. If you had the situation where we were dealing with complex numbers-- and if you don't know what a complex number is, or an imaginary number, don't worry too much about it. But if you were dealing with those, then you would have to keep the absolute value of x there. Because then this would be defined for numbers that are less than 0.
Video transcript
- [Voiceover] So we have here the square root, the principal root, of one two-hundredth. And what I want to do is simplify this. When I say "simplify it" I really mean, if there's any perfect squares here that I can factor out to take it out from under the radical. And so I encourage you to pause the video and see if you can do that. Alright so there's a couple of ways that you could approach this. One way is to say, Well this is going to be the same thing as the square root of one over the square root of 200. The square root of one is just one over the square root of 200. And there's a couple of ways to try to simplify the square root of 200. I'll do it a couple of ways here. Square root of 200. You could realize that, OK, look 100 is a perfect square. And it goes into 200. So this is the same thing as two times 100. And so the square root of 200 is the square root of two times 100, which is the same thing as the square root of two times the square root of 100. And we know that the square root of 100 is 10. So it's the square root of two times 10 or we could write this as 10 square roots of two. That's one way to approach it. But if it didn't jump out at you immediately that you have this large perfect square that is a factor of 200, you could just start with small numbers. You could say, alright, let me do this alternate method in a different color. You could say, ah it's the same color that I've been doing before. (laughing) You could say that the square root of 200, say Well it's divisible by two. So it's two times 100. And if 100 didn't jump out at you as a perfect square, you could say, Well that's just going to be two times 50. Well I can still divide two into that. That's two times 25. Let's see, and 25, if that doesn't jump out at you as a perfect square, you could say that that's not divisible by two, not divisible by three, four, but it is divisible by five. That is five times five. And to identify the perfect squares you would say, Alright, are there any factors where I have at least two of them? Well I have two times two here. And I also have five times five here. So I can rewrite the square root of 200 as being equal to the square root of two times two. Let me just write it all out. Actually I think I'm going to run out of space. So the square root, give myself more space under the radical, square root of two times two times five times five times two. And I wrote it in this order so you can see the perfect squares here. Well this is going to be the same thing as the square root of two times two. This second method is a little bit more monotonous, but hopefully you see that it works, (laughing) I guess is one way to think about it. And they really, they boil down to the same method. We're still going to get to the same answer. So square root of two times two times the square root times the square root of five times five, times the square root of two. Well the square root of two times two is just going to be, this is just two. Square root of five times five, well that's just going to be five. So you have two times five times the square root of two, which is 10 times the square root of two. So this right over here, square root of 200, we can rewrite as 10 square roots of two. So this is going to be equal to one over 10 square roots of two. Now some people don't like having a radical in the denominator and if you wanted to get rid of that, you could multiply both the numerator and the denominator by the square root of two. 'Cause notice we're just multiplying by one, we're expressing one as square root of two over square root of two, and then what that does is we rewrite this as the square root of two over 10 times the square root of two times the square root of two. Well the square root of two times the square root of two is just going to be two. So it's going to be 10 times two which is 20. So it could also be written like that. So hopefully you found that helpful. In fact, even this one, you could write if you want to visualize it slightly differently, you could view it as one twentieth times the square root of two. So these are all the same thing.