Hey everyone. Early in the course when we studied exponents, we saw how to square a number and we saw something like 4 squared was equal to 16. But now what's going to happen in problems is they'll give you the right side of the equation, like 16, and they're going to ask you for the left side. They're going to ask you what number, when I multiply it by itself, gets me to 16. And to answer this question, we're going to talk about square roots. Now you've probably seen square roots at some point in math classes before, but we're going to 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 going to be 1? Well, no. Because 1 multiplied by itself is 1. What about 2? Now that just gets me 4. What about 3? 3, 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 2 square roots, 3 and negative 3. This actually always works for positive real numbers. They always have 2 roots. There is a positive root like the 3, and textbooks sometimes call that the principal roots, but there's also the negative roots, the negative 3. Alright? So, basically, if I start at 9 and I want to go backwards and take the square roots, there are 2 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 how there's a problem here. So if there are 2 possible answers for the square root of 9
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 9 . 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 roots, 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 a 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 them at the same time. So it's a little bit 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 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 squared is not going to be that because that's just 1. 2 squared is 4. 3 squared is 9. 4 squared 4 times 4 is 16. So I have to keep going. I got 5 squared, which is 25. That's still not it. And what about 6 squared? Well, 6 squared is equal to 36. So it meansTable of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
0. Review of Algebra
Radical Expressions
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