Common Core Algebra I.Unit 9.Lesson 1.Square Roots

Alone, welcome to another common core algebra one lesson. By E math instruction. My name is Kirk Weiler, and today we're going to be doing unit 9 lesson number one on square roots. Before we begin, let me just remind you that you can find a worksheet for this video and a homework assignment by clicking on the video's description or by visiting our website at WWW dot E math instruction dot com. Also, don't forget about the QR codes at the top of every one of our worksheets. Use your phone or your tablet to scan that code and you'll be taken to the video that goes with that lesson. Okay, let's begin. I'm hopeful that square roots are things that you've seen before. All right? There are certainly something that have come up before in mathematics. And today we're going to look at them. Starting off with sort of, if you will, easy square root problems. Exercise one says find the value of each of the following principal square roots, write a reason for your answer in terms of a multiplication equation. Now this is a good opportunity for you to see how much you remember about square roots. I understand the fractions and letter E and letter F might be a little bit daunting. So if those are a problem, I wouldn't be overly concerned about it. But pause the video now and see if you can evaluate each one of the following square roots. Each one of them has only one answer. So don't write down two, even though you might think that's correct. Pause the video now. All right, let's go through them. So square roots always answer a very funny question. They always answer the following question. What number squared will give me 25? And the answer there is 5. All right? Why? This is the right of reason. Well, that's because 5 times 5 or better yet, 5 squared is equal to 25. Square roots and squaring are inverses of each other. Their opposites of each other. Okay? So the square root of 9, well, that's three. Why? Because three times three is equal to 9. The square root of hundred, well, that's equal to ten. Why? Because ten times ten is equal to a hundred. The square root of zero is kind of a funny one. I'll have even students who get all the way to pre calculus that are confused by this. But it shouldn't be confusing. The square root of zero is zero. Why? Because zero times zero is equal to zero. Now, just as a reminder, about fractions. Don't worry about these two for a minute. When you multiply fractions together, let's say one third times actually, let me get rid of that. One third times two fifths. What you always do is you multiply the numerators two times one is two. And you multiply the denominators three times 5 is 15. We can use that to understand that the square root of one fourth must be one half, and the reason why is that one half times one half is one fourth, right? One times one is one, two times two is four. All right? Likewise, the square root of 64 9 has to be 8 thirds. Why? Because 8 thirds times 8 thirds, 8 times 8 is 64. Three times three is 9. Okay? So square roots essentially answer the question what number do I have to multiply by itself in other words, what number do I have to square to get the number under the square root? The number under the square root under the square root has a special name. It's called the radicand. The radic hand. That's not a term that a ton of teachers will use, but some of them will. And it's convenient as opposed to saying things like the number under the square root. So I'll sometimes say the number of the square root and sometimes I'll say the ratic hand. Okay? Anyhow, write down anything you need to pause the video and then we'll move on. All right, here we go. Let's go to the next page. Now, there are two square roots of every positive number. And the reason for that is because the product of two negatives is positive, right? So we know that a negative and a negative, negative times a negative is a positive, and the product of two positives. Is it positive? And because of that, every positive integer has two square roots. You know, for example, right? Negative three times negative three is 9, and three times three is 9. So both positive three and negative three are legitimate square roots of 9. Likewise, negative one 5th times negative one 5th, this one 25th, and one 5th times one 5th is one 25th. So both of those are legitimate square roots of one 25th. So you might want to ask, or you might ask, well, why didn't I put down two answers then for every problem in number one? Well, that's because we need to know. We need to have a way of knowing whether we're talking about the positive square root of a number, or the negative square root of a number. All right? So which square root we're talking about will be dictated simply by whether the square root has a negative attached to it. In other words, the square root of 9 is three. But the negative square root of 9 is negative three. All right, so when you see the square root symbol, and you see no negative in front of it, then we're talking about what's known as the principle. Square root. And yeah, that is principle is in the guy who sits in the corner office. The guy or the woman who sits in the corner office, I apologize. And there really shouldn't be a space there. But it's called the principle square root. And then there's the negative square root. So unless otherwise stated, you figure out which one you're talking about by whether there's a negative or a positive sitting in front of the square root, or visually speaking a negative or not a negative. All right. Let me clear out the text. And then let's move on and do some more problems with square roots. All right. Exercise two, this should be a piece of cake. Give all square roots of each of the following numbers. All right? Pause the video now and go ahead and do that. Well, as we just discussed, each number has two square roots, and you may have just written them down, but let me just emphasize. One of them will be the positive square root of four, which is two. And one of them will be the negative square root of four, which is negative two. In this problem, one of them will be the positive square root of 36, which is 6. And one of them will be the negative square root of 36, which is negative 6. Here one of them will be the positive square root of one 16th, which will be one fourth, and one of them will be the negative square root of one 16th. Which will be negative one fourth. And again, if you had all of these just written down without the square root symbols, that's fine, but I just really want to emphasize that that's how we're going to mark the difference between the positives and the negatives. All right. I'm going to clear this out, continuing on with our pretty simple square root work until the second half of the lesson. All right, here it goes. Let's move on to the next problem, okay? So exercise three is a relatively simple multiple choice question, but it as usual involves some function notation, right? F of X equals the square root of X plus three. Which of the following is the value of F of 46, right? So again, this is more than anything else about function notation, right? And just knowing that when you see that, you shouldn't be scared. You should just put a 46 in. In this case, anywhere there's an X so we then add these numbers first, we get the square root of 49, and that's out. All right. So simple enough. Okay, you just have to understand function notation and what the square root symbol means. But it's still important, right? That can be tricky, especially the function notation. So I'm going to clear out the text right down anything you need to. All right. And let's keep moving on. Okay? Now, the next problem might seem a little silly to you. Okay? But its purpose is to illustrate one of the most important properties that square roots have. All right? And that's a property that has something to do with products. So let's do a and B together. Then I'm going to have you do C and D on your own. Exercise forces find the value of each of the following products, right? So this thing says, hey, find the product of the square root of four times the square root of 9. Well, the square root of four is two. The square root of 9 is three, so that product is equal to 6. Letter B though says the square root of four times 9. Well, that implies that we should do the product of four times 9 first, which is 36, and then we should find the square root of 36. But that's 6, right? Oh, huh. I wonder if that's a coincidence, right? I mean, it almost appears that when we're multiplying two square roots together, we could multiply the radicals first, the numbers under the square root, and then take the square root second, or we could take the square roots first, and then multiply. Anyway, why don't you do problems C and D to see if the pattern continues to hold. All right, let's do it. While the square root of four times the square root of 25, the square root of four is two. Square root of 25 is 5 and two times 5 is ten. On the other hand, if we multiply the ratic hands first, four times 25, we get 100. And the square root of a hundred is ten. So yeah, the pattern continues to hold, right? And this illustrates, as I said, one of the most important properties that square roots have. Which is that square roots, if you will, respect multiplication. Or play nice with multiplication, however you want to think about it. Anyway, we're going to formally look at it in just a minute, but I'm going to clear out this text. So pause the video now, write down anything you need to, and then we're clearing it out. All right, let's move on to the next page. So the multiplication property of square root says, if we're going to multiply two square roots together, we can do that by first multiplying the numbers under the square root, the two radicans, and then take the square root. Strangely enough, we're going to use this property almost more in the opposite order. That is, if we've got the product of two numbers underneath the square root, then we can break up that product as the product of the two individual square roots. All right? So that's how we're going to use this property. At least mostly. But we will do a little bit of it using sort of that the first version of it here. All right? So I'm going to clear this out, and then we're going to practice on some problems that involve this property. Okay? So one of the immediate immediate things that we can do with this property is multiply unfriendly square roots, right? They're friendly square roots like the square root of four and the square root of 9 and the square root of 25. And then there are unfriendly ones, like the square root of two and the square root of 8. But our property says, look, if I've got the square root of two times the square root of 8, then I can actually break that into the square root of two times 8, which is the square root of 16, and that's four. That's it. Now, of course, we could have multiplied those two numbers together and gotten a number under the square root that wasn't a perfect square, 16 is perfect square. But it turns out that in these problems that happens a lot. So the square root of 12 times the square root of three will be the same as the square root of 12 times three. 12 times three is 36, and the square root of 36 is 6. All right, looks good. How about letter C? Why don't you pause the video and do that one on your own? All right, well hopefully you got ten is the final answer. Because just like before, we can actually multiply the radicans first. 20 times 5 is 100. And then the square root of a hundred is ten. Okay? So one of the immediate consequences of the multiplication property of square roots is that if you've got two square roots multiplying each other, it may be easiest to multiply the radicans first and then take the square root. Maybe. All right. So I'm going to clear this out. And then we're going to talk about the other major consequence of that property. And that is in simplifying square roots. Now what does it mean to simplify square root? What it means is to find an equivalent expression, so a number that's identical. But where all the perfect square factors have been removed from the erratic hand. All right. Now, this is actually a procedure that you're going to learn that is specifically mandated if you're taking math or common core algebra in New York State. If you're taking common core algebra in other states, it may not be something that you're required to learn at the common core level. Or the common core algebra one level that is. But let's jump into it. It's kind of a, it's kind of a fun process. It's not too hard. And let's see how it works. Okay? So we're going to do it with one specific example. Exercise number 6 says letter a, list out the first ten perfect squares. Now, perfect squares are numbers that have nice square roots, like one squared is one. So that's a perfect square. Two squared is four. So that's a perfect square. Three squared is 9. So that's a perfect square. Four squared is 16. So 16 is a perfect square. 5 squared is 25. So that's a perfect square. 6 squared is 36. Perfect square. 7 squared is 49. Perfect square. 8 squared is 64. Perfect square. 9 squared is 81. Perfect square. And ten squared is a hundred. Now there's nothing magical about the first ten. In fact, you might have to go past that. You might have to go to 11 squared, 12 squared, 13 squared, but it's pretty rare. So now let's play around with letter B it says now consider the square root of 18. So 18 is not a perfect square, right? Look at that. One, four, 9, 16, 25, 36, et cetera. 18 is not one of them. But one of those, actually, two of those, but let's eliminate the first one. I really don't care about this one. All right, one of those divides 18 nicely. And that is 9. Right? Specifically 18 equals 9 times two. So watch what we're going to do now. We're now going to reverse this process, right? The square root of 18 is the same as the square root of 9 times two. Because obviously, 9 times two is 18. So that's true no matter what. I wouldn't matter what property we were using. But now that property about square roots says, well, we can rewrite this as the square root of 9 times the square root of two. The square root of 9, we can do something with. We can say it's three. Square root of two, we can't do anything with, so we have to leave it as the square root of two. And that's it. The square root of 18 is the same as three times the square root of two, and they really are the same. If you take your calculator out and you type in square root of 18, enter. And then you type in three times the square root of two, enter. You will get exactly the same and ugly, decimal expression. Okay? So let's clear this out. And then get some more practice with simplifying square roots. The key buzzword here, sorry, I should have pointed this out before, was simplest, radical form, simplest radical form. Anyway, I'm going to clear this out. So copy down what you need to. All right, here we go. Let's move on. Okay, at the top of the page what I did was I listed out the first 12 perfect squares and really the one that is the least relevant is the number one. We're never really going to worry about the fact that that's a perfect square. So it's not that it's not perfect square. I just don't really want to care about it. So it says write out each of the following square roots and simplest radical form. So here's what we want to do. We want to find the biggest one of these that divides into 8 nicely. And that's four. All right? So in other words, 8 is the same as four times two. But then we can use that property about square roots to rewrite that as the square root of four times the square root of two, and then we can change the square root of four into two. So take the square root of what you can and leave what you can't. All right? So I can find the square root of four is two. But the square root of two, I have to leave that way. All right, let's take a look at the square root of 45. We think about these numbers, which number goes in. They're obviously you don't have to go up any higher than 45. But the biggest one, biggest perfect square that goes into 45 is 9, right? 45 is 9 times 5. Which means I can now break this up as the square root of 9 times the square root of 5 always put the perfect square first. The square root of 9, I can find that's three. The square root of 5, I can't find, so I just leave it like that. All right, letter C, I want to show you something that's kind of cool. All right, because it's often a mistake that students will make. The look at the square root of 48 and very, very legitimately. They'll have their calculator out and they'll say, okay, square root of 48 divided by four. And they'll find out, hey, you know, 48 is divisible by four, right? So 48 is actually four times 12. Okay, so they'll break it up as the square root of four times the square root of 12. Square root of four is two, and then they'll leave it like this. Now there is no question, by the way, that the square root of 48 is two times the square root of 12. We've done everything that legitimate here. The problem is that 12 also has a perfect square sitting in it. Right? Let me get rid of this. Right. So we can actually take the square root of 12 and break it into the square root of four times the square root of three. But then that'll be two times two Times Square root of three. Which will be four root three. Now, there's nothing wrong with that. That's my final answer. That is simplest radical form. This amazingly important phrase simplest radical form. But if I did it in two steps, that means there's actually a bigger perfect square that divides nicely into 48. In fact, the bigger one that divides into it nicely is 16. So square root of 48 is the same as the square root of 16 times three. So that's the square root of 16 times the square root of three and square root of 16 is four. And square root of three is square root of three. All right. Now there is absolutely nothing nothing nothing nothing wrong. With reducing it like that. That's sort of the equivalent if you had to reduce 12 eighteenths, let's say. And you said, oh, I know two goes into both of them. So you did this, and you got 6 ninths, and then you said, oh, wait a second. No, I don't know. I'm going to divide that by three in that by three. And get two thirds. That would be fine. You know, you can do that in two steps, or you can do it in one step by dividing both numerator and denominator by 6. I don't want that right now. Let's go back to this. All right, now there's a negative in front of this one. That's no big deal. We just have to not lose it. So we look at all of these problems. And we say, all right, not problems, but we look at all those numbers, and we say, well, the biggest one of those that goes into 75 is 25. So I'm now going to break that down into negative root 25 times root three. Don't put two negatives attached there, then you'd have a positive. The square root of 25, I can reduce negative 5, the square root of three. I can't do anything with so negative 5 Times Square root of 30. Times Square root of three. All right. What I'd like you to do is play around with E and F on your own. Pause the video, take up to 5 minutes, calculator maybe even handy to help you figure out which perfect squares go into those numbers. And then see what you get. All right, let's go through them. So the largest perfect square that goes into 72 happens to be 36. In fact, 36 times two is 72. So I break this end of the square root of 36 and the square root of two, the square root of 36 6, and I leave the square root of two. All right, the square root of negative 500. I'm sorry. That was bad. The negative square root of 500 is not the square root of negative 500. We'll talk about that more next time. Well, the biggest perfect square that goes into 500 is 100. So I can then break that down as negative square root of a hundred. Times Square root of 5, and then that will be negative ten. Times root 5. All right. And that's how we simplify a square root. All right, by extracting all the perfect square factors. All right, so pause the video now and write down anything you need to. This is going to be an important skill for us in future problems. All right, here we go. Let's finish up the lesson. So today we saw a lot of different things. We reviewed the basic idea of the square root. We learned how square roots of any positive numbers always kind of come in pairs. And we learned how to distinguish between the principle square root. That's the positive one, and the negative square root by simply whether there was a negative in front of the radical or not. Then we saw this important property of square roots, the product property, right? That allowed us to find the product of two square roots, but it also allowed us to simplify square roots by pulling out their perfect square factors. All right. For now, though, I'd like to thank you for joining me for another common core algebra one lesson by E math instruction. My name is Kirk Weiler, and until next time, keep thinking. And keep solving problems.