All right, everybody. Let's dive back into the text. We are going to be looking at investigating roots, but I ordered our notes for the second round of notes. Square roots, cube roots, and ordering real numbers. This is a number symbol. But I use that because when you're taking notes, if you abbreviate things that you know what the abbreviations mean, it can help you with taking notes. All right. So, and this is was today is uh for you. For me, it's still 8:28, but anyway, there's the date. We got our pages. We got our title. Much better than, you know, notes. And so let's take a look at it. So I can evaluate square roots and cube roots. That's what we're going to do. And let's start out with the build your understanding. Okay? And I'll show you what to write down, but let's read it together. Aaron is also planning to make an or origami cubes of different sizes. The edge lengths of the cubes are 1 in, 2 in, 3 in, 4 in, and 4 in. So we can see then that's what he wants to make. Uh talk to a partner. Okay. So let's let's make this table in our notes. So we want to put it' be nice if I could draw cubes here, but let's just say or uh I put origami or origami. Sorry about that. Okay. So, let's make our I just wanted to misspell it so that you felt better. You know, if you ever misspel anything, we can always fix it. All right. Edge length inches and volume in inches cubed. Okay. So the volume okay so the volume for a cube and in this in my notes here my abbreviation V stands for volume but instead of writing the whole word volume I just write V and to know what shape I'm talking about I put the name of the shape right there and this is what I do for all volumes so the volume of a cube is the side length to the third power. S stands for the side length. All right. So that's what we're doing. So if the side length or the edge length, same thing. The edge length is 1 in, then the volume is going to be 1 to the 3, which is also one 1 in cubed. We don't even have to write the inch cubed though because it is given an inch cub. So I'm not going to write it for the next one. If the edge length is two, the volume would be 2 to the 3 power, which is 2 * 2 * 2, which is not 6, that's 2 + 2 + 2, it's 8. So the volume of the edge length is 2, the volume is eight. three 3 cubed is 27 and 4 cubed is 64. So now let's read the questions. What do you notice about how the volume changes as the edge length changes as the So now if they ask us for to look at patterns then we can we can analyze the data. I'm just going to write it like this. And the first thing I would do to analyze this data is I'm just going to look at the uh the interval change. So on the left side I'm increasing by one each time. And I want to just investigate how much am I increasing by over here. And I actually not even sure but I'm just looking. So I see it increases by seven and this is increase by what is that 19 and then in order to find the increase we just subtract. So that's seven and then I had to borrow. So that's 37. I don't really see anything there. But what I do notice is that the volume is the edge cubed. I don't know if that's what they're looking for. If you are given the volume volume of a cube, how can you find the edge length? Now, I do think it's what they're looking for. So, this didn't really get me anywhere, but I wanted to try it to see if it worked. So my answer to C is going to be the volume is the edge length times itself three times. for the edge length cubed. Same thing. So D, it says if you are given the volume of a cube, how can you find the edge length? The volume of a cube as I already said is the edge length cubed. So maybe I don't know. I don't know if it uh wanted me to do something different in the table, but anyway. Now, the most important question on the page is E right here. Let's read it. What is the edge length of a cube with a volume of 125 cubic inches? Why is it difficult to express the edge length of a cube with a volume of 10 cubic in? Like I said, that's that's important. So if the volume of the cube is 125, then we want to ask ourself what number times itself three times equals 125. And it has to be the same number. So we can't do like 1* 1* 125. It has to be because that would be different numbers. Maybe you already know it's five * 5 * 5. So that means that for if we wanted to continue this table down, we could add five and 125 would be the uh volume of a cube whose edge length is pi. Now the second question is uh if the volume of a cube is 10, what is the edge length? So it's like that. What's the edge length if the volume is 10? And the reason that's difficult is because what number times itself three times is 10? then it's got to be the same number. Well, the answer is it's an irrational number. So, the only thing that would work is an irrational number which we talked about can't even be written. You can't only write those things in um in uh symbols. And so, I'm going to show you the symbol for that. It would look like this. the cube root of 10. That's the only thing that would work. You see see what I wrote there? It's it looks like a square root sign or a long division sign with a little three up there and then the 10. And I just wrote that three times and that's what would equal 10. Okay. So now that leads us into our vocabulary. So we're talking about square roots and cube roots. And so I'm going to pause the video and write down my vocabulary and then you can also have it in your notes. I'll be back. All right, I got all my vocab feel pretty good. Square root. The square root of p is x when x^2 = p. For example, the square root of 9 = 3 because 3^2 = 9. 3 * 3 is 9. In perfect squares, a perfect square is a whole number whose square roots are integers. Okay, look at that on the next page. Radical symbol. The symbol, this guy right here, this little housey symbol is used to indicate a positive or principal square root. And you can leave some I left some space and wrote the next one. But we are going to look at all right. So use information. Find the square roots of 81. Okay. So that's the first thing we want to find the square roots of 81. My eight. There we go. 81. So what number? So what number would that would be nine because 9 squared is 81. So we ask ourselves what squared is that number? Well the answer is nine. Okay but it says it gives us two spaces. So there is another answer. What else squared is 81? The answer is a -9. 9 * -9 is also 81. So we also have to say -9. But the principal square root would be the positive number. Okay. Then also we have the the fraction. Some people get confused with the fraction. So what about four 9ths? How do we let me write that the square root of 4 9ths? What we do is we take the square root of the top and the square root of the bottom separately. That's one way. And the square root of four is two. 2 * 2 is 4. And we already know the square of 9 is 3. So the square of 4 9ths is 2/3. Doing good. All right. And so that's just um Okay. Suppose Aaron wants a square piece of origami paper with an area of two square cm. What is the length of one side of the piece of paper? Use a square root symbol in your answer. And so if Aaron wants the area of a square, so he wants the area of a square that's two, right? So that that means like so what could the side lengths be so that they multiply to two? Well, like we talked about it' be the square of 2 * the 2 and that's two. But we can't really think of a number a whole a whole number or even a decimal where that's true. So that's why we have to write it as a symbol. And we'll just note that the square root of two is irrational. Okay, crazy daisy. Irrational numbers. Talk more about them, but first we have to talk about cube roots. Okay, so we're going to do the same exercise, but this time with cube roots. So let's write down 8 27. Okay, so we want to find the cube root of 8 27ths. So we're going to find the cube root of 8 and the cube root of 27. So what number to the third power? What number to the third power is 8? Or what number times itself three times is 8? I think earlier in the video we said that's two. 2 * 2 * 2 is 8. And I bet you can figure out that 3 * 3 is 9 and and 9 * 3 is 27. So 27 is three because three 3es make a 27 or 3 to the 3 power is 27. All right. And so we're going to do fact for understanding in uh class. But we have one more problem to do here. We might have to solve something. And this is example four. This is example three. This is two. Okay. So let's say we have to solve x2 = 100. In order to solve that, we're going to square root both sides and we get x = the of 100, which is 10. And as we learned earlier, it could also be -10. Both of those numbers can be put in for x, and then we get 100. All right, one more page. I'm thinking we'll do three pages of notes each week, which I think is like more than fair. Okay, so the last vocabulary word is real numbers. Real numbers are numbers on the number line on the uh the number line, you know, that are rational or irrational. So, we've learned about rational numbers. Now, we just learned about irrational numbers. Both of those put together make up the real number system. So with our squares and our cubes, what's going to make these things really doable for us is having a list of the perfect ones. Be easier to make a list of perfect people because there are none, right? But there are perfect squares. They're very arrogant little fellows. Okay. So, uh we'll say 1 4 because 2 * 2 is 4. 3 * 3 is 9. 16 25. I'm just multiplying by myself in my head. And I'm going to go up to 15, 81, 100, 121, 144. Uh 1 15 * 5 uh that's 12 oh 169 196 225 and we might as well do 16 * 16 296 I think 17 * 17 is 343 but I could be wrong. All right. So, you can go up as high as you want, but at least to 15. In the perfect cube list, we're going to go up 10 spaces on that. So, one, and we already talked about 2 cubed is 8. 3 cubed is 27. 4 cubed is 64. 5 cubed is 125. 6 cubed is 216. 7 cubed is 3. That's the 343. 8 cubed is 512. 9 cubed is 729 and 10 cubed is 1,000. You see I got those memorized. Why? Because it helps me when I have to estimate. So in the second section of our week, we have to estimate the square root of 50. So let's write that down. Estimate square root of 50. Well, if you look, 50 is not on our list, but 49 is. So, if I want to estimate the square root of 50, I'm going to surround it by perfect squares. Surround it by the close closest perfect squares. So, 49 is on the lower end. And what is on the higher end? 64. All right. And I even use my inequality signs. So the square root of 49 is 7. The square root of 64 is 8. So I know that 50, the square of 50 is going to be a number in between 7 and 8. And as I look at this using my analysis that I tried earlier, I see that 50 is only one away from 49 and it is like 14 away from 64. So when I think about this number, first of all, I know it's going to be 7 something. But when I think about the decimal place, I'm going to choose a really low tth value because it's really close to the lower end. So I'm going to make it 7.1. Now if I have to do the square root of 63 using that same analysis, that's way closer to 64. So I might say 7.9. Second And then let's say I have the square root of something in the middle of 49 and 64, maybe 56. So that's that's pretty much in the middle of these two, right? That's like 7 more than 49 and 8 less than 64. So I might approximate that as 7.5. So we're using a little bit of judgment. The most important thing is to get the whole number right and then have a reasonable approximation for the 10th value. We can do the same thing with cube roots. But look seeing the similar thing right. Estimate the cube root of 100 to the nearest 100th. Yeah, that is something. 100 has a perfect square, but it doesn't have a perfect cube. So, let's look in the list. 100 is between 64 and 125. So, let's do that. So, we're going to do 64 cube root. And then we got the 100, which we don't know yet. And then we have 125. So the cube root of 64 in my list is 1 2 3 4. It's four. And the cube root of 125 is 1 2 3 4 5. So I know that the cube root of 100 is going to be between four and five. And so what will it be? Well, let me do the analysis. How much bigger is 164? If I subtract, it's 36 bigger. And then it's 25 bigger. So, you know, it's close to the middle. So, I might say 4 maybe it's a little bit closer to the higher end. So, I could say 4.6 4.5. I'm going to say 4.6 six because it's closer. And then, you know, we could check it on a calculator. So, the estimation of the cube root of 100 is 4.6. I lied. We're going to do another line and be done. Just having too much fun around here. Okay. So, let's see where these things would be on actually Okay, let's do 3 a and then we'll call it a day. Okay, you can use estimates of square roots to help you estimate and compare numerical expressions involving square roots. Compare the values below. Write less than or greater than to complete each statement. All right, let's write our statements in. So this is three on page 354. Okay. So two circle six. So 3 3 + 2 circle 3 + 6. Okay. So we want to use some less than and greater than signs. So 2 is less than 6. So the of 3 + 2 is less than the 3 + 6. Does that make sense? We're trying to build our number sense. -1 -3. Okay. Now be careful here. -1 is greater than -3. Okay? It's not less than because the negatives go backwards. So the square so the square of 7 minus one is greater than the square of 7 - 3. And that makes sense, right? That was a good one. I'm glad we did that one. Okay. Six. Oh goodness. 67. Six is less than seven. So there two times the of 6. That's 2 * 6 is less than 2 * the 7. So when we just put them next to each other, that's multiplication. And let's do a negative -5 is is greater than -7. So I had to think about that one for a second. So -5 * the 3 is greater than -7 * the of 3. All right. See, you really got to think. So we will do B, C, and D in class or maybe even on a test and and four as well. And then guess what guys? After you've done that and we do these practice problems, see the rest of these are just practice problems. And then what is that? Review after two weeks. And notice look a vocabulary. All these vocabulary we have done rational, irrational, repeating decimals, cubes, squares. We did it all. We just did the whole chapter 10, module 10 in two weeks. And next up is the Pythagorean theorem for our next video. But first, we're gonna review and uh test and everything. And you are going to do great. All right, I will see you there.