Hey everyone. Early in the course when we studied exponents, we saw how to square a number and we saw something like \(4^2\) was equal to 16. But now what's gonna happen in problems is they'll give you the right side of the equation, like 16, and they're gonna ask you for the left side. They're gonna ask you what number, when I multiply it by itself, gets me to 16. And to answer this question, we're gonna talk about square roots. Now, you've probably seen square roots at some point in math classes before, but we're gonna go over it again because there are a few things that you should know. Let's go ahead and take a look. So, basically, the idea is that squares and square roots are like opposites of each other. The reverse of squaring a number is taking the square root. So, for example, if I were asked what are the square roots of 9, I have to think of a number. When I multiply it by itself, it gets me to 9. So let's try that. Is it gonna be 1? Well, no. Because 1 multiplied by itself is 1. What about 2? Now, that just gets me 4. What about 3? \(3^2\), if I multiply it by itself, you know, square it over here, I get to 9. But is that the only number that works for? Well, actually, no. Because remember that negative 3, if I square negative 3, the negative sign cancels, and I also just get to 9. So, in other words, there are two numbers that when I multiply them by themselves, they get me to 9. And what that means is that 9 has two square roots, 3 and negative 3. This actually always works for positive real numbers. They always have two roots. There is a positive root like the 3, and textbooks sometimes call that the principal roots, but there's also a negative root, the negative 3. Alright? So, basically, if I start at 9 and I want to go backwards and take the square roots, there are two possible solutions. I have 3 and negative 3. So how do we write that? Well, we use this little radical symbol over here, this little, this little symbol. And so if I go backwards from 9, I get to 3 or negative 3. But notice there's a problem here. So if there are two possible answers for the square root of 9, how do I know which one I'm talking about? Am I talking about 3 or negative 3? Because sometimes in problems, you'll just see a square root like this. How do you know which one it's talking about? Basically, it comes down to the way that you write the notation. So what we do here is the radical symbol when it's written by itself, that means it's talking about the positive root. So if you just see radical 9 by itself, it's just talking about the positive root of 3. And to talk about the negative root, you have to stick a minus sign in front of that radical symbol. That means that now you're talking about the negative root, which is the negative 3. Alright? So it's super important that you do that, because what I learned when I was studying this stuff is that if you just have radical 9, you could sort of just write plus or minus 3, but you can't do that. This is incorrect. And if you try to do this, you actually write this on homework or something like that, you may get the wrong answer. Right? So just be very, very careful. The notation is very important here. Alright? And then what you also see sometimes is that if you want to talk about both of these at the same time, you'll see a little plus or minus in front of the radical. That just means that you're talking about plus and minus 3. So both of more efficient that way. Alright. So that's all there is to it. So let's just actually go ahead and take a look at our first two problems here. If I want to evaluate this radical, I have radical 36. So, in other words, I need to take the square root of 36, and I need a number that multiplies by itself to get me 36. So let's just try. \(1^2\) is not going to be that because that's just 1. \(2^2\) is 4. \(3^2\) is 9. \(4^2\) 4 times 4 is 16. So I have to keep going. \(5^2\), which is 25. That's still not it. And what about \(6^2\)? Well, \(6^2\) is equal to 36. So it means all of these are wrong answers, but this one's the right one. I have 6. When I multiply by itself, it gets me 36. So which one is it but that also means that negative 36 oh, I'm sorry. Negative 6 also gets me to 36. So what's the answer here? Is it the positive or is it the negative? Remember, the radical is by itself, so this actually means it's just talking about 6, and it's not talking about both of them or the negative one. So it's very important. What about the second one here? Now we see a negative that's in front of the radical symbol. That means it's talking about the negative root of 36. So this answer is negative 6 mplified by itself, gets you negative 36 or, you know, in this case, negative 9. Can I do that? Well, here, what happens is if I try to do 3, remember \(3^2\) is not negative 9. It's just positive 9. So that's not gonna work. And what about negative 3? That's also not gonna work because if I took negative 3 and squared it, I, you know, I just got a positive 9. So, in other words, that's not gonna work either because that just equals 9. So how do I take the square root of a negative number? It turns out you just can't do it. You can't do it because no matter what number you pick, when you multiply it by itself, the negative just cancels out. And so what happens is all you need to know for right now is that whenever you see a negative that's inside of a radical, you just need to know that it's imaginary. And we'll cover this later on, but that's all you need to know for now. So here's a good sort of like memory tool to use. When we saw negatives that were outside of radicals, that was perfectly fine, and that was okay. So, for example, we saw negative outside of radical 9 or 36. That's perfectly fine. But if you see a negative inside, that means that it's imaginary. So outside is okay, but inside is imaginary. Alright? So negative radical 36 over here, perfectly fine, but the radical of negative 36, that's imaginary. Alright? That's all you need to know for now. Anyway, folks, so that's all there is to it. Let me know if you have any questions. Thanks for watching.