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Square roots and real numbers (old)

An old video of Sal where he simplifies square roots in order to determine whether they represent rational or irrational numbers. Created by Sal Khan and CK-12 Foundation.
Video transcript
I have here a bunch of radical expressions, or square root expressions. And what I'm going to do is go through all of them and simplify them. And we'll talk about whether these are rational or irrational numbers. So let's start with A. A is equal to the square root of 25. Well that's the same thing as the square root of 5 times 5, which is a clearly going to be 5. We're focusing on the positive square root here. Now let's do B. B I'll do in a different color, for the principal root, when we say positive square root. B, we have the square root of 24. So what you want to do, is you want to get the prime factorization of this number right here. So 24, let's do its prime factorization. This is 2 times 12. 12 is 2 times 6. 6 is 2 times 3. So the square root of 24, this is the same thing as the square root of 2 times 2 times 2 times 3. That's the same thing as 24. Well, we see here, we have one perfect square right there. So we could rewrite this. This is the same thing as the square root of 2 times 2 times the square root of 2 times 3. Now this is clearly 2. This is the square root of 4. The square root of 4 is 2. And then this we can't simplify anymore. We don't see two numbers multiplied by itself here. So this is going to be times the square root of 6. Or we could even right this as the square root of 2 times the square root of 3. Now I said I would talk about whether things are rational or not. This is rational. This part A can be expressed as the ratio of 2 integers. Namely 5/1. This is rational. This is irrational. I'm not going to prove it in this video. But anything that is the product of irrational numbers. And the square root of any prime number is irrational. I'm not proving it here. This is the square root of 2 times the square root of 3. That's what the square root of 6 is. And that's what makes this irrational. I cannot express this as any type of fraction. I can't express this as some integer over some other integer like I did there. And I'm not proving it here. I'm just giving you a little bit of practice. And a quicker way to do this. You could say, hey, 4 goes into this. 4 is a perfect square. Let me take a 4 out. This is 4 times 6. The square root of 4 is 2, leave the 6 in, and you would have gotten the 2 square roots of 6. Which you will get the hang of it eventually, but I want to do it systematically first. Let's do part C. Square root of 20. Once again, 20 is 2 times 10, which is 2 times 5. So this is the same thing as the square root of 2 times 2, right, times 5. Now, the square root of 2 times 2, that's clearly just going to be 2. It's going to be the square root of this times square root of that. 2 times the square root of 5. And once again, you could probably do that in your head with a little practice. The square root of the 20 is 4 times 5. The square root of 4 is 2. You leave the 5 in the radical. So let's do part D. We have to do the square root of 200. Same process. Let's take the prime factors of it. So it's 2 times 100, which is 2 times 50, which is 2 times 25, which is 5 times 5. So this right here, we can rewrite it. Let me scroll to the right a little bit. This is equal to the square root of 2 times 2 times 2 times 5 times 5. Well we have one perfect square there, and we have another perfect square there. So if I just want to write out all the steps, this would be the square root of 2 times 2 times the square root of 2 times the square root of 5 times 5. The square root of 2 times 2 is 2. The square root of 2 is just the square root of 2. The square root of 5 times 5, that's the square root of 25, that's just going to be 5. So you can rearrange these. 2 times 5 is 10. 10 square roots of 2. And once again, this is irrational. You can't express it as a fraction with an integer and a numerator and the denominator. And if you were to actually try to express this number, it will just keep going on and on and on, and never repeating. Well let's do part E. The square root of 2000. I'll do it down here. Part E, the square root of 2000. Same exact process that we've been doing so far. Let's do the prime factorization. That is 2 times 1000, which is 2 times 500, which is 2 times 250, which is 2 times 125, which is 5 times 25, which is 5 times 5. And we're done. So this is going to be equal to the square root of 2 times 2-- I'll put it in parentheses-- 2 times 2, times 2 times 2, times 2 times 2, times 5 times 5, times 5 times 5, right? We have 1, 2, 3, 4, 2's, and then 3, 5's, times 5. Now what is this going to be equal to? Well, one thing you might see is, hey, I could write this as, this is a 4, this is a 4. So we're going to have a 4 repeated. And so this the same thing as the square root of 4 times 4 times the square root of 5 times 5 times the square root of 5. So this right here is obviously 4. This right here is 5. And then times the square root of 5. So 4 times 5 is 20 square roots of 5. And once again, this is irrational. Well, let's do F. The square root of 1/4, which we can view this is the same thing as the square root of 1 over the square root of 4, which is equal to 1/2. Which is clearly rational. It can be expressed as a fraction. So that's clearly rational. Part G is the square root of 9/4. Same logic. This is equal to the square root of 9 over the square root of 4, which is equal to 3/2. Let's do part H. The square root of 0.16. Now you could do this in your head if you immediately recognize that, gee, if I multiply 0.4 times 0.4, I'll get this. But I'll show you a more systematic way of doing it, if that wasn't obvious to you. So this is the same thing as the square root of 16/100, right? That's what 0.16 is. So this is equal to the square root of 16 over the square root of 100, which is equal to 4/10, which is equal to 0.4. Let's do a couple more like that. OK. Part I was the square root of 0.1, which is equal to the square root of 1/10, which is equal to the square root of 1 over the square root of 10, which is equal to 1 over-- now, the square root of 10-- 10 is just 2 times 5. So that doesn't really help us much. So that's just the square root of 10 like that. A lot of math teachers don't like you leaving that radical in the denominator. But I can already tell you that this is irrational. You'll just keep getting numbers. You can try it on your calculator, and it will never repeat. Your calculator will just give you an approximation. Because in order to give the exact value, you'd have to have an infinite number of digits. But if you wanted to rationalize this, just to show you. If you want to get rid of the radical in the denominator, you can multiply this times the square root of 10 over the square root of 10, right? This is just 1. So you get the square root of 10/10. These are equivalent statements, but both of them are irrational. You take an irrational number, divide it by 10, you still have an irrational number. Let's do J. We have the square root of 0.01. This is the same thing as the square root of 1/100. Which is equal to the square root of 1 over the square root of 100, which is equal to 1/10, or 0.1. Clearly once again this is rational. It's being written as a fraction. This one up here was also rational. It can be written expressed as a fraction.