 Okay, in this video, I'm going to talk about the properties of square roots, specifically two prop. Ah, darn it, I did it again. You know what? I'll just trim it off. Okay, in this video, I'm going to talk about the properties of square roots. Now, I'm not going to talk necessarily what a square root is. By now, in Algebra 2, we should know this. Off to the right side, though, I do have reminders. The square root of 1 is 1, the square root of 4 is 2, so on and so forth. You can see those on the right side. What I'm going to talk about today is the product property and the quotient property. These two properties we use to simplify square roots when we can't take the square root of a number. Okay, so what I'm going to do is I'm going to show this just with number examples, and I know in previous videos I've done examples with variables and with numbers. I'm going to skip the variables on this one. I'm just going to do examples with numbers. Okay, so the product property. Products means multiply. If I want to take a number and I want to take a square root and reduce it, I can actually use multiplication to help me with this. For example, if I have the square root of 50, and now if I look at my list over here, I don't know what the square root of 50 is. I get down here to the square root of 49, which is 7, so I know it's something close to 7, but I don't know exactly what it is. What I can do is I can simplify this and get this down to a nicer, neater number other than just the square root of 50 by using the product property. Okay, so what I'm going to do is I'm going to take 50, and this is what I commonly call take 50 and split it up, okay, split it up using multiplication. So again, product property, split this up using multiplication. So what I'm going to do is instead of writing this as 50, I can rewrite this as something different. I'm going to rewrite this as square root symbol 25 times 2, notice that little dot there, times 2. 25 times 2 is the same thing as 50, so I haven't really changed much of everything. Yes, the numbers look different, but it's still 50 underneath there. Now why would we split it up this way? The reason we do that is because if you look at this list over here, I actually know what the square root of 25 is, okay? So square root of 50 changed into 25 times 2. That square root of 25, I actually know what that is. So that is why we split it up using product, using multiplication, and so that I can simplify it down to something a little bit smaller, okay? Now what I can do with this is I'm going to actually separate these two numbers. I'm going to take 25, the square root of 25 times the square root of 2. This doesn't look a lot different, but notice now I have two radicals, two square root symbols, and now that I've kind of separated them, now I can actually evaluate the square root of these numbers. The square root of 25, I actually know what that is, that is 5, but then the square root of 2, I actually don't know what that is. So if I get down to a number that I don't know, I'm just going to leave it. So this is, square root of 25 is 5, square root of 2 is just going to stay the square root of 2. Now normally when we write this, that would be it, now we're done, I can't simplify that anymore. 5 is as good as it gets, it's out of the radical, I don't need to mess with it anymore. The square root of 2, I don't know what the square root of 2 is, it's this long irrational number, I'm not going to mess with that, I'm not going to split that up, nothing like that. So this is it, this is as good as it gets. I can rewrite this as simply 5 square root of 2. You can call it 5 square root of 2, 5 times the square root of 2, I like to commonly refer that to this as 5 root of 2, just kind of a simpler way to say it. So that's an example of the product property, a very simple example. Now I'm going to do an example with the quotient property. Product means multiply, quotient is then going to mean divide. So if I have, if I'm dividing something, this is how it's going to simplify out to. So for example, if I have 36 over 4, and I'm taking the square root of that fraction. Okay, now there's actually a couple different ways to simplify a flat fraction like this, but I'm just going to demonstrate this using the quotient property that we know. I'm hopefully going to do a couple of different examples to kind of help us out with this. Okay, so 36 over 4, what I can do, and this is kind of similar to what I do with the product property, I'm going to take these and split these up. So instead of taking the square root of 36 divided by 4, 36 fourths, what I'm going to do is I'm going to change this, so I'm taking the square root of 36 on top and the square root of 4 on bottom. So instead of taking the square roots altogether of those two numbers, I'm actually individually taking the square roots of them, very similar to what I did up here. Instead of doing the square root of everything, I'm going to do the square root of the two individual numbers. And instead of taking the square root of everything, I'm going to take the square root of the two individual numbers. Now recognize that 36 and 4, we can take the square root of both of those numbers. So notice that this is going to reduce, the square root of 36 is 6, and the square root of 4 is 2, 6 divided by 2 that reduces down even farther, and that is going to be a simple number of 3. Okay, now there is another way you could have done this, some of you might have already seen this, 36 divided by 4 is 9, and so the square root of 9 is 3. That's kind of another way to do that, but the way that I did this demonstrated that I can separate the two numbers. Another example that I can use would be something similar to, let's use, let's just use a simple example, let's do the square root of 18 over the square root of 2, make a big division symbol there, make a big fraction bar there. Okay, so this is a separate example. Now if you look at both of these, notice that the square root of 18, I don't know what that is. The square root of 2, I don't know what that is, so it makes no sense to separate these two numbers. In the previous problem, we separated both of them, we were able to evaluate the square root, and everything worked out. In this case, it's not that. We don't know what the square root of 18 is, we don't know what the square root of 2 is. So what I can do is I can actually bring these back together. So instead of having two separate square roots, I now have one square root with both numbers underneath that radical, underneath that square root symbol. Now the reason that we do this is now it's actually going to be easier for us to evaluate the square root if I actually figure out what the fraction is first. 18 divided by 2 is 9. So sometimes it's actually easier to bring the two numbers back together underneath the radical, figure out what the fraction is, and then reduce. So in this case, we actually got the same answer as what we got above, which is 3. So the quotient rule, there's kind of two ways you can look at it. You can think of it as either take a fraction that's underneath the radical and split it up, take the square root of the top, square root of the bottom, or you can think of it as, well, if I'm taking the square root of the top, square root of the bottom, I can go backwards and bring them back together underneath a single radical. Couple of different ways to look at it. So those are the two properties, those are two of the properties of square roots, the product property and the quotient property.