Hi everyone and welcome to our lecture on square roots. So we've seen square roots before. I think the first time when we were studying area back in the day we're studying geometry. So for example, to find the area of a square, I think we all know that the formula for the area of square is area equals side square. So well let s denote the length of our side. Here's my beautiful square with all sides equal and of course right angles everywhere. So A will be the total area and s is the length of a side of this. So now imagine if the side of the square is say 3 centimeters long, then the area of a square is 9 centimeters. The calculation is not hard. We take 3 squared, we get 9. Of course these are units. So we get 9 square centimeters or 9 centimeters squared. So conversely, you can use the formula to find the length s of a side of a square, if we're given it's area. So if the area of a square is 25 centimeters squared, then the side squared is 25. And of course if we solve for s, we could argue, well, s is either 5 or negative 5. Now, each of these solutions technically is a square root of 25. However, but because lengths are always not negative and we're measuring a distance, 5 is the only possible solution that we're going to care about it. It's a little weird to come back and say the length is negative 5 centimeters. And this is going to introduce our first definition. We're going to call the positive solution of the square root, our principal square root. So if A is any non negative number, so it's allowed to be greater than or equal to 0. The principal square root of A denoted by the square root sign of A is the non negative number b such that b squared equals A. When we talk about square roots, when we use this symbol, the square root symbol here, we always want something that's non negative. So for example, why do I keep saying non-negative? Why can I just say positive? Well, as some examples that we saw, principal square root or just the square to 25, this is going to be positive 5. It's important realize that if I use this symbol, I really do want the positive number. Think of it from a geometric standpoint. I only want the positive value. If I said what is the square to 16? We do not say positive or negative. We just say positive 4. Now, here's one more for you. What do you think the square of 0 is? Can you think of a non negative number b such that b squared is 0? What number squared is equal to 0? Well, of course that's equal to 0. So I'm restricting the numbers that I put under the square root to be non negative. You can have positive, you can have 0. You just can't have negative numbers underneath. That leads into the world of imaginary or complex numbers and we'll deal with that another time. Not every square root that you put under this symbol has to be a nice square root that we know. There's nothing wrong with talking about the square root of 2. This is the principal square root of 2. Now, this is not a nice number that we probably have memorized, but it is a number, if you throw in the calculator, you can certainly get that this is approximately 1.41421 and it goes on forever and ever and ever. So some of these are a little nicer than other, but they're all just perfectly fine real numbers. And the square roots are the subject of this lecture in the section in the course. Let's just do a couple of examples. What is the square root of 144? And pause the video. See if you know this, or you don't know this. Hopefully this is just 12. We've had this drilled into us since when we were child. Let's do a second version of this. If s squared is equal to 144, then solve for s, ready? In this case, since I'm going to take the square double size, I want all solutions to this. I'm going to write that s as plus or minus 12. Now notice, it wasn't asked of me with the square symbol. So I'm allowed to get back both the positive and negative answer. To see this a different way, if I start with X squared equals 144, I can algebraically subtract 144 from both sides. Hopefully you agree. If I subtract 144 from both sides, I have s squared minus 144 0. Then I can factor this quadratic. I have s minus 12 in parentheses. S plus 12. Stare at that for a second, foil this back together, convince yourself that this is true. Now I have two things multiplied that give me 0. So of course I have S minus 12 is 0 and S plus 12 is 0. I write both of them equal to 0, which gives of course the solutions s is positive 12 or s equals negative 12. More concise way of writing that of course is s is plus or minus 12. The algebraic way to factor is obviously a little longer, but it really shows you what's going on. It's good to see it to understand what's happening. So if you have a quadratic, you should expect to solutions back and there's the plus or minus. If you start with the square root, they only want the principal one. Then you return just the positive number or he positive principle square root. One more, just to practice with fractions as well. If I said to you, what is the square root of 4 over 9, what number squared gives me four nineths? Again, this is just the positive number because I'm introducing the root symbol, that's what that symbol means. This would just be two thirds. Now if I said to you, if x squared is equal to four ninths, solve for X. Let's just do the shortcut. Let's just take the square to both sides. I'll say X equals plus or minus the square root of four ninths. Notice when I introduce the square root symbol I must put plus or minus, is often a source of confusion for a lot of students. And of course now x equals plus or minus two ninths. You can always tell how many answers you're going to get back based on the kind of question you're starting with. If the square doesn't appear in the question, they're just looking for the quadratic solutions, the two solutions. If the square does appear in the questions like these first examples here, then you're looking for just the principal value of the square root, the non negative number. So now let's talk about other routes. So we've seen the positive solution to s squared equals 25 is denoted by square root of 25. Similarly, the positive solutions to something like s the fourth equals 25 is denoted as the fourth root of 25. In general, if n is even the positive solution, to x to the n is some number, let's say 25 as well, is denoted as the nth root of 25. That's how you read this. You read this as the nth route. When you're talking about roots in general, the number n is called the index. And just note, when you write square roots, n equals 2 is implied. So when there's no number that appears, it's assumed to be a square root, it is not written. It's assumed that the index is 2. See a simple example of this. If I said to you, find x such that x cubed is equal to negative 8. What I'm looking for is the cubed root of negative 8. If I want the positive solution, so what number times itself 3 times is negative 8. What times what times what is negative 8. In general would say that the cube root of negative 8 is in fact negative 2. So in general, the odd route of a negative number is a negative number. This is okay to have a answer here be a negative number. Notice I'm no longer restricting myself to the principal square root because I've left the world the square roots and now we're talking about cube roots. Another example, what would the fourth root of 16 be? Again without a calculator I have to ask you nice numbers, but this just understand the symbols and the notation. What times what times what times what is 16? What number raised to the fourth is 16? Just remember, behind these routes, I'm asking you to solve the equation what number raised to the fourth is 16. Can you figure it out? Well, of course, in this case this is equal to 2, positive 2. One more, just to play around with it. What if I said negative 16. What's the forth root of negative 16? To understand what this expression is asking, write it as an equation and say what number raised the fourth is negative 16? In this particular case, I can't take a real number, raise it to itself four times. Of course because if I have positive, I'm going to get positive, I have negative four, negative is going to be positive. I'm never going to get a number when raised to the fourth is going to give me negative, never. So in this particular case we'd say no real solutions. This will have complex solutions, but we'll save that for another time. If we're restricted to real numbers, which we usually are, we'll say no real solutions. Some square roots are rational numbers. We saw before that the square to 16 is 4. It's very nice. The square root of 4, of course, is 2. Maybe you can even talk about the square to 1 being 1. Some square roots are just nice rational numbers. So friendly reminder, a rational number is one that I can write as a ratio of integers, right? As a fraction. But some other square roots are just not nice numbers that you're not going to solve these, you're going to need a calculator to do these. For example, the square root of 2 we saw this is a nice irrational number. Now, this is a big claim that I make that there's no fraction. I can't find a fraction A over B such that if I squared is equal 2. To appreciate this for a minute, you sort of taking this on faith a little bit, but I'd like to walk through the process. It's a nice example of sort of logical deduction or in direct reasoning. Talk about why this doesn't exist. This is a classic example. So once again, what am I going to show here? I'm going to show that route 2 is irrational. And sort of ask the question like, why is it irrational? And I'm going to assume that I can write it as a fraction a over b squared and just remember what I'm allowed to do a over b. They're integers. Of course B is not 0. I want to divide by 0. So I just have some integers and I'm going to assume that the fraction a over b is reduced. That's a fancy way to say no common terms. So I don't want it to be like two forths, I'm going to think of it as one half. They have no common factor, all right? So let's start with my equation here. A over b squared equals 2. And what I'm going to do is I'm going to bring that exponents into both the numerator and the denominator. That's all fine. And I'll move things around and I'll say a squared is 2b squared. If a square does 2b squared, then that means that a squared is 2 times some number. So of course a squared is even. And you can show do you think about this. If a squared is even if some numbers squared is even, then a itself has to be even. The only way to get an even number in its square, like if I start with 4, I have to start with 2. You can't take an odd number like 3, 5 or 7squared. You always get an odd number back. Do you think about why that's true? But that turns out to be true. So a is in particular even and from that I'm going to write a as 2 times some other number. Let's just call it 2 times t. And we'll put it back into our equation. So we have 2t, where this is the expression for a is an even number squared is 2b squared. Let's bring the 2 in and I get 4t squared is 2b squared. Divide both sides by 2. And I get 2t squared equals b squared. And all of a sudden from here I could see that b squared is equal to 2 times some number. So guess what? B squared is also even. And for the same reasoning as above, if b squared is even then b is also even. Now I claim we have a problem because if a is even and b is even and I'm thinking of a fraction a over b. And remember, I said from the very beginning, we have no common terms. I've just contradicted myself, I've arrived at a contradiction. So when we say this, we say a over b is not reduced, you have an even number of an even number like 6 over 4. You can certainly reduce it. And this is a flaw in my original assumption that there was some fraction square that equal 2. In particular, this means that the square root of 2 is not rational. Cannot write the square of 2 as a over b. You can use a similar argument to show that 3 is irrational or in general root n is irrational wherein is some whole number, but just not the square of another whole number. This is a nice little proof and I wanted to show you guys one of the arguments that you can do to prove things and walk through some of the reasoning behind why square roots are not rational. It turns out, you can write the square root of x as equal to x to the one half. Let's think about why this is true. If I take both of these numbers and square them. So if we square both sides, I get X has to be equal o x to the one half squared. But of course when I have an exponent of an exponent, this just turns out to be x the one which is in fact x. So we're going to write sometimes our roots with rational exponents. The square root would be written as x to the one half. The cube root of x will be written as x to the one third. The fourth root of x will be written as x to the one forth and so on. In general the nth root of x is written as x to the one over n. These routes have nice properties because they're related to the properties of exponents. So for example, if I have x to the y to the one over n, I can bring in the exponent to both pieces. So I have x to the one over n times y to the one over n. But if I convert all of that to square roots or n root in this case, I know that the nth root of xy is going to be equal to the n root of x times the nth root of y. So roots distribute over multiplication, for all the same reasons. If I had a fraction x over y o the one over n, I can bring this n as x to the one over n, to the y one over n. But just using similar, but different notation that says that if I have the nth root of any fraction, I can bring the n root in. This is true for squares as well, which is the most common case that will probably work with, but you can bring the n root in over a fraction or a product. One more property that we're going to use comes from the algebraic fact that, if I have an exponent 2 an exponent, I get to multiply the exponents together. In particular, if I have one of them be a nth route, what if I had X to the one over n raised to the s. Well, this becomes x to the n over s. So therefore, if I replace x over n with it nth route, something like this. I can bring the exponent inside right on the s. This is going to allow us to simplify and write our exponents using roots. Whenever you want to simplify roots, maybe it's easier just to convert it to exponents and work it out. But the big three properties that I want you to come away with is that roots distribute over multiplication, roots distribute over fractions, and I'll have powers of a nth root. I can move that power right inside. Let's do some examples for a second. So 16 to the 1/4. I see this as a fraction again. Let's try to do this without the calculator. This would be the forth root of 16. Unless we know what number to itself. 4 times is equal to 16. Well this is of course 2. Let's do another one. What if I had 16 to the 5/4 is more complicated, but I want you to see this as 16 to the 1/4 raised to the 5. So break up that fraction. Focus on the denominator, which is really acting as forth root and then raises to the 5 and all of a sudden the fourth was 16. Well, that we know that's just 2 and raising 2 to the 5. Now, that's much simpler. You can check that is in fact 32. So 16 raised, the 5/4th is just 32. Let's do another one. How about 125 raised to the negative 4/3? How do you solve this? How do you think about this without using a calculator? Now that negative, don't let it scare you. Remember one way to think about this is throw this in the denominator and answer the same question now with a positive number, but just work inside the denominator. Let's once again take the same approach as before. We'll just keep working downstairs. 125 to the 1/3 raised to the 4 is of course the cube root of 125 raised to the fourth. Again, all still in the denominator. What times, what times, what is 1 25? What number raised to itself? What number times itself? 3 times? Well, that's just 5. So really this question is asking what is 1/5 to the fourth? And you can check that's 625. So our final answer is 1 over 625. Again, we're in the denominator because the original exponents was in fact negative. Let's do one more, see if I can trick you hear. What is negative 16 raised to the 1/4? Pause the video for a second, see if you can solve this. Remember, this is asking for what number raised the fourth is negative 16? Well, let's just say there's no real solutions. So it is okay when working with exponents, when working with roots to come back and say sorry, I can't get you an answer. There's no real solutions here. If you want to get fancy you can start going to the world complex numbers, but we'll save that for another day. Let's do one more. What if I had 2 route 3 equals root x? Solve for x. Again, pause the video, see if you can solve it. Let's think about this for a second. How would I solve for x? X is trapped under a square root. So why don't I square both sides to get rid of it? So let's see. So we have square both sides. We get 2 route 3 squared is equal to squared of x square. That's of course just x. Now, let's bring the 2 in to both pieces. We get 4 times 3 or 3 times. Root 3 times root 3 is just 3 and that's going to equal x. So x of course is equal to 12. Let's go back and actually check that just to see why that's in fact true. If I look at the square to 12 by the rules of exponents, I can think of this as the square to 4 times 3 because 4 splits his 4 times 3. But now I can break this up as the square to 4 times the square of 3. Remember, roots split over multiplication. Square to 4, well, hey, that's just 2. And I'm left with, as promised 2 or 3, the thing I started off with. So you can use these as software numbers. You can use these software equations. We're going to see them all over the place. So practice these practices and again, try not to rush over the calculator, try to understand how the exponents is manipulating the root or vice versa. And how to solve and understand what these exponents are actually doing. All right, great job on this video. We'll see you next time.
