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Dividing positive and negative numbers Pre-Algebra Khan Academy.mp3 | The times symbol, people don't want to confuse it with the letter x, which gets used a lot in algebra. And so that's why they use the dot very often. So this just says negative 7 times 3 in the numerator. And then we're going to take that product and divide it by negative 1. So in the numerator, negative 7 times 3. Positive 7 times 3 would be 21. But since exactly one of these two are negative, this is going to be negative 21, and it's going to be negative 21 over negative 1. |
Dividing positive and negative numbers Pre-Algebra Khan Academy.mp3 | And then we're going to take that product and divide it by negative 1. So in the numerator, negative 7 times 3. Positive 7 times 3 would be 21. But since exactly one of these two are negative, this is going to be negative 21, and it's going to be negative 21 over negative 1. And so negative 21 divided by negative 1. Negative divided by a negative is going to be a positive. So this is just going to be positive 21. |
Dividing positive and negative numbers Pre-Algebra Khan Academy.mp3 | But since exactly one of these two are negative, this is going to be negative 21, and it's going to be negative 21 over negative 1. And so negative 21 divided by negative 1. Negative divided by a negative is going to be a positive. So this is just going to be positive 21. Let me write all these things down. So if I were to take a positive divided by a negative, that's going to give me a negative. If I have a negative divided by a positive, that's also going to give me a negative. |
Dividing positive and negative numbers Pre-Algebra Khan Academy.mp3 | So this is just going to be positive 21. Let me write all these things down. So if I were to take a positive divided by a negative, that's going to give me a negative. If I have a negative divided by a positive, that's also going to give me a negative. If I have a negative divided by a negative, that's going to give me a positive. And if I have, obviously, a positive divided by a positive, that's also going to give me a positive. Now let's do this last one over here. |
Dividing positive and negative numbers Pre-Algebra Khan Academy.mp3 | If I have a negative divided by a positive, that's also going to give me a negative. If I have a negative divided by a negative, that's going to give me a positive. And if I have, obviously, a positive divided by a positive, that's also going to give me a positive. Now let's do this last one over here. This actually is all multiplication, but it's interesting because we're multiplying three things, which we haven't done yet. And we could just go from left to right over here, and we could first think about negative 2 times negative 7. Negative 2 times negative 7, they are both negative. |
Dividing positive and negative numbers Pre-Algebra Khan Academy.mp3 | Now let's do this last one over here. This actually is all multiplication, but it's interesting because we're multiplying three things, which we haven't done yet. And we could just go from left to right over here, and we could first think about negative 2 times negative 7. Negative 2 times negative 7, they are both negative. The negatives cancel out, so this will give us, this part right over here, will give us positive 14. And so we're going to multiply the positive 14 times this negative 1. Times negative 1. |
Dividing positive and negative numbers Pre-Algebra Khan Academy.mp3 | Negative 2 times negative 7, they are both negative. The negatives cancel out, so this will give us, this part right over here, will give us positive 14. And so we're going to multiply the positive 14 times this negative 1. Times negative 1. Now we have a positive times a negative. Exactly one of them is negative, so this is going to give me a negative answer. It's going to give me negative 14. |
Dividing positive and negative numbers Pre-Algebra Khan Academy.mp3 | Times negative 1. Now we have a positive times a negative. Exactly one of them is negative, so this is going to give me a negative answer. It's going to give me negative 14. Now let me give you a couple more, I guess we could call these trick problems. What would happen if I had 0 divided by negative 5? Well, this is 0 negative fifths, so 0 divided by anything that's non-zero is just going to be equal to 0. |
Dividing positive and negative numbers Pre-Algebra Khan Academy.mp3 | It's going to give me negative 14. Now let me give you a couple more, I guess we could call these trick problems. What would happen if I had 0 divided by negative 5? Well, this is 0 negative fifths, so 0 divided by anything that's non-zero is just going to be equal to 0. What if we were to do it the other way around? What happens if we said negative 5 divided by 0? Well, we don't know what happens when you divide things by 0. |
Dividing positive and negative numbers Pre-Algebra Khan Academy.mp3 | Well, this is 0 negative fifths, so 0 divided by anything that's non-zero is just going to be equal to 0. What if we were to do it the other way around? What happens if we said negative 5 divided by 0? Well, we don't know what happens when you divide things by 0. We haven't defined that. There's arguments for multiple ways to conceptualize this, so we traditionally just say that this is undefined. We haven't defined what happens when something is divided by 0. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | Let me just do a quick graph of these just so we can visualize what they look like. So let me draw a quick graph over here. So our first point is 7, negative 1. So 1, 2, 3, 4, 5, 6, 7. This is the x-axis. 7, negative 1. So 7, negative 1 is right over there. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | So 1, 2, 3, 4, 5, 6, 7. This is the x-axis. 7, negative 1. So 7, negative 1 is right over there. 7, negative 1. This of course is the y-axis. And then the next point is negative 3, negative 1. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | So 7, negative 1 is right over there. 7, negative 1. This of course is the y-axis. And then the next point is negative 3, negative 1. So we go back 3 in the horizontal direction. Negative 3. But the y-coordinate is still negative 1. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | And then the next point is negative 3, negative 1. So we go back 3 in the horizontal direction. Negative 3. But the y-coordinate is still negative 1. It's still negative 1. So the line that connects these two points will look like this. It will look like that. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | But the y-coordinate is still negative 1. It's still negative 1. So the line that connects these two points will look like this. It will look like that. Now, they're asking us to find the slope of the line that goes through the ordered pairs. Find the slope of this line. And just to give a little bit of intuition here, slope is a measure of a line's inclination. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | It will look like that. Now, they're asking us to find the slope of the line that goes through the ordered pairs. Find the slope of this line. And just to give a little bit of intuition here, slope is a measure of a line's inclination. And the way that it's defined, slope is defined as rise over run, or change in y over change in x, or sometimes you'll see it defined as the variable m, and then they'll define change in y as just being the second y-coordinate minus the first y-coordinate, and then the change in x as the second x-coordinate minus the first x-coordinate. These are all different variations in slope, but hopefully you appreciate that these are measuring inclination. If I rise a ton when I run a little bit, if I move a little bit in the x-direction and I rise a bunch, then I have a very steep line. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | And just to give a little bit of intuition here, slope is a measure of a line's inclination. And the way that it's defined, slope is defined as rise over run, or change in y over change in x, or sometimes you'll see it defined as the variable m, and then they'll define change in y as just being the second y-coordinate minus the first y-coordinate, and then the change in x as the second x-coordinate minus the first x-coordinate. These are all different variations in slope, but hopefully you appreciate that these are measuring inclination. If I rise a ton when I run a little bit, if I move a little bit in the x-direction and I rise a bunch, then I have a very steep line. I have a very steep upward sloping line. If I don't change at all when I run a bit, then I have a very low slope, and that's actually what's happening here. I'm going from, you could either view this as the starting point or view this as the starting point, but let's view this as the starting point. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | If I rise a ton when I run a little bit, if I move a little bit in the x-direction and I rise a bunch, then I have a very steep line. I have a very steep upward sloping line. If I don't change at all when I run a bit, then I have a very low slope, and that's actually what's happening here. I'm going from, you could either view this as the starting point or view this as the starting point, but let's view this as the starting point. So this negative 3 comma 1. If I go from negative 3 comma negative 1 to 7 comma negative 1, I'm running a good bit. I'm going from negative 3, my x value is negative 3 here, and it goes all the way to 7. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | I'm going from, you could either view this as the starting point or view this as the starting point, but let's view this as the starting point. So this negative 3 comma 1. If I go from negative 3 comma negative 1 to 7 comma negative 1, I'm running a good bit. I'm going from negative 3, my x value is negative 3 here, and it goes all the way to 7. So my change in x here is 10. To go from negative 3 to 7, I change my x value by 10. But what's my change in y? |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | I'm going from negative 3, my x value is negative 3 here, and it goes all the way to 7. So my change in x here is 10. To go from negative 3 to 7, I change my x value by 10. But what's my change in y? Well, my y value here is negative 1, and my y value over here is still negative 1. So my change in y is 0. My change in y is going to be 0. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | But what's my change in y? Well, my y value here is negative 1, and my y value over here is still negative 1. So my change in y is 0. My change in y is going to be 0. My y value does not change no matter how much I change my x value. So the slope here is going to be, when we run 10, what was our rise? How much did we change in y? |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | My change in y is going to be 0. My y value does not change no matter how much I change my x value. So the slope here is going to be, when we run 10, what was our rise? How much did we change in y? Well, we didn't rise at all. We didn't go up or down. So the slope here is 0. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | How much did we change in y? Well, we didn't rise at all. We didn't go up or down. So the slope here is 0. Or another way to think about it is this line has no inclination. It's a completely flat, it's a completely horizontal line. So this should make sense. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | So the slope here is 0. Or another way to think about it is this line has no inclination. It's a completely flat, it's a completely horizontal line. So this should make sense. This is a 0. The slope here is 0. And just to make sure that this gels with all of these other formulas that you might know, but I want to make it very clear, these are all just telling you rise over run, or change in y over change in x, a way to measure inclination. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | So this should make sense. This is a 0. The slope here is 0. And just to make sure that this gels with all of these other formulas that you might know, but I want to make it very clear, these are all just telling you rise over run, or change in y over change in x, a way to measure inclination. But let's just apply them just so hopefully it all makes sense to you. So we could also say slope is change in y over change in x. If we take this to be our start, and if we take this to be our end point, then we would call this over here x1, and then this is over here, this is y1, and then we would call this x2, and we would call this y2. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | And just to make sure that this gels with all of these other formulas that you might know, but I want to make it very clear, these are all just telling you rise over run, or change in y over change in x, a way to measure inclination. But let's just apply them just so hopefully it all makes sense to you. So we could also say slope is change in y over change in x. If we take this to be our start, and if we take this to be our end point, then we would call this over here x1, and then this is over here, this is y1, and then we would call this x2, and we would call this y2. If this is our start point and that is our end point. And so the slope here, the change in y, y2 minus y1, so it's negative 1 minus negative 1. All of that over x2, negative 3 minus x1, minus 7. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | If we take this to be our start, and if we take this to be our end point, then we would call this over here x1, and then this is over here, this is y1, and then we would call this x2, and we would call this y2. If this is our start point and that is our end point. And so the slope here, the change in y, y2 minus y1, so it's negative 1 minus negative 1. All of that over x2, negative 3 minus x1, minus 7. So the numerator, negative 1 minus negative 1, that's the same thing as negative 1 plus 1. And our denominator is negative 3 minus 7, which is negative 10. So once again, negative 1 plus 1 is 0 over negative 10. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | All of that over x2, negative 3 minus x1, minus 7. So the numerator, negative 1 minus negative 1, that's the same thing as negative 1 plus 1. And our denominator is negative 3 minus 7, which is negative 10. So once again, negative 1 plus 1 is 0 over negative 10. And this is still going to be 0. And the only reason why we have a negative 10 here and a positive 10 there is because we swapped the starting and the ending points. In this example right over here, we took this as the start point and made this coordinate over here as the end point. |
Slope from two ordered pairs example 2 Algebra I Khan Academy.mp3 | So once again, negative 1 plus 1 is 0 over negative 10. And this is still going to be 0. And the only reason why we have a negative 10 here and a positive 10 there is because we swapped the starting and the ending points. In this example right over here, we took this as the start point and made this coordinate over here as the end point. Over here we swapped them around. 7, negative 1 was our start point, and negative 3, negative 1 is our end point. So if we start over here, our change in x is going to be negative 10, but our change in y is still going to be 0. |
Find measure of supplementary angles Angles and intersecting lines Geometry Khan Academy.mp3 | We're told that the measure of angle QPR, so QPR, so that's this angle right over here, is 2x plus 122. And I'll assume that these are in degrees. So it's 2x plus 122 degrees. And the measure of angle RPS, so that's this angle right over here, is 2x plus 22 degrees. And they ask us to find the measure of angle RPS. So we need to figure out this right over here. So we would be able to figure that out if we just knew what x is. |
Find measure of supplementary angles Angles and intersecting lines Geometry Khan Academy.mp3 | And the measure of angle RPS, so that's this angle right over here, is 2x plus 22 degrees. And they ask us to find the measure of angle RPS. So we need to figure out this right over here. So we would be able to figure that out if we just knew what x is. And lucky for us, we can use the information given to solve for x and then figure out what 2 times x plus 22 is. And the big idea here, the thing that pops out here, is that the outside rays for both of these angles form a line. These two angles form a line. |
Find measure of supplementary angles Angles and intersecting lines Geometry Khan Academy.mp3 | So we would be able to figure that out if we just knew what x is. And lucky for us, we can use the information given to solve for x and then figure out what 2 times x plus 22 is. And the big idea here, the thing that pops out here, is that the outside rays for both of these angles form a line. These two angles form a line. You could say that they are supplementary. Those of these angles are supplementary. 2x plus 22 plus another 2x plus 122 is going to add up to 180. |
Find measure of supplementary angles Angles and intersecting lines Geometry Khan Academy.mp3 | These two angles form a line. You could say that they are supplementary. Those of these angles are supplementary. 2x plus 22 plus another 2x plus 122 is going to add up to 180. We know that this entire angle right over here is 180 degrees. So we can say that the measure of angle QPR, this angle right over here, 2x plus 122, plus the green angle, plus angle RPS, so plus 2x plus 22, is going to be equal to 180 degrees. Is going to be equal to 180. |
Find measure of supplementary angles Angles and intersecting lines Geometry Khan Academy.mp3 | 2x plus 22 plus another 2x plus 122 is going to add up to 180. We know that this entire angle right over here is 180 degrees. So we can say that the measure of angle QPR, this angle right over here, 2x plus 122, plus the green angle, plus angle RPS, so plus 2x plus 22, is going to be equal to 180 degrees. Is going to be equal to 180. And now we can start simplifying this. We have 2x's. We have another 2x's. |
Find measure of supplementary angles Angles and intersecting lines Geometry Khan Academy.mp3 | Is going to be equal to 180. And now we can start simplifying this. We have 2x's. We have another 2x's. So those are going to add up to be 4x. And then we have 122 plus 22. So that's going to be 144. |
Find measure of supplementary angles Angles and intersecting lines Geometry Khan Academy.mp3 | We have another 2x's. So those are going to add up to be 4x. And then we have 122 plus 22. So that's going to be 144. And the sum of those two are going to be equal to 180 degrees. We can subtract 144 from both sides. On the left-hand side, we're just going to be left with a 4x, this 4x right here. |
Find measure of supplementary angles Angles and intersecting lines Geometry Khan Academy.mp3 | So that's going to be 144. And the sum of those two are going to be equal to 180 degrees. We can subtract 144 from both sides. On the left-hand side, we're just going to be left with a 4x, this 4x right here. And on the right-hand side, we're going to have, let's see, if we were subtracting 140, we would have 40 left. And then we have to subtract another 4. So it's going to be 36. |
Find measure of supplementary angles Angles and intersecting lines Geometry Khan Academy.mp3 | On the left-hand side, we're just going to be left with a 4x, this 4x right here. And on the right-hand side, we're going to have, let's see, if we were subtracting 140, we would have 40 left. And then we have to subtract another 4. So it's going to be 36. Divide both sides by 4. And we get x is equal to 9. Now remember, we're not done yet. |
Find measure of supplementary angles Angles and intersecting lines Geometry Khan Academy.mp3 | So it's going to be 36. Divide both sides by 4. And we get x is equal to 9. Now remember, we're not done yet. They didn't say solve for x. They said find the measure of angle RPS, which is 2 times x plus 22, or 2 times 9 plus 22, which is 18 plus 22, which is equal to 40. So the measure of angle RPS is 40 degrees. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | And remember, we're going to do parentheses first, parentheses, p for parentheses, then exponents. Don't worry if you don't know what exponents are, because this has no exponents in them. Then you're going to do multiplication and division. They're at the same level. Then you do addition and subtraction. So some people remember PEMDAS. But if you remember PEMDAS, remember multiplication, division, same level, addition and subtraction, also at the same level. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | They're at the same level. Then you do addition and subtraction. So some people remember PEMDAS. But if you remember PEMDAS, remember multiplication, division, same level, addition and subtraction, also at the same level. So let's figure what order of operations say that this should evaluate to. So the first thing we're going to do is our parentheses, and we have a lot of parentheses here. We have this expression in parentheses right there, and then even within that, we have these parentheses. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | But if you remember PEMDAS, remember multiplication, division, same level, addition and subtraction, also at the same level. So let's figure what order of operations say that this should evaluate to. So the first thing we're going to do is our parentheses, and we have a lot of parentheses here. We have this expression in parentheses right there, and then even within that, we have these parentheses. So our order of operations say, look, do your parentheses first, but in order to evaluate this outer parentheses, this orange thing, we're going to have to evaluate this thing in yellow right there. So let's evaluate this whole thing. So how can we simplify it? |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | We have this expression in parentheses right there, and then even within that, we have these parentheses. So our order of operations say, look, do your parentheses first, but in order to evaluate this outer parentheses, this orange thing, we're going to have to evaluate this thing in yellow right there. So let's evaluate this whole thing. So how can we simplify it? Well, if we look at just inside of it, the first thing we want to do is simplify the parentheses inside the parentheses. So you see this 5 minus 2 right there. We're going to do that first no matter what, and that's easy to evaluate. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | So how can we simplify it? Well, if we look at just inside of it, the first thing we want to do is simplify the parentheses inside the parentheses. So you see this 5 minus 2 right there. We're going to do that first no matter what, and that's easy to evaluate. 5 minus 2 is 3. And so this simplifies to, I'll do it step by step. Once you get the hang of it, you can do multiple steps at once. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | We're going to do that first no matter what, and that's easy to evaluate. 5 minus 2 is 3. And so this simplifies to, I'll do it step by step. Once you get the hang of it, you can do multiple steps at once. So this is going to be 7 plus 3 times the 5 minus 2, which is 3. And then you have, and all of those have parentheses around it. And of course, you have all this stuff on either side, the divide for, no, whoops, that's not what I want. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | Once you get the hang of it, you can do multiple steps at once. So this is going to be 7 plus 3 times the 5 minus 2, which is 3. And then you have, and all of those have parentheses around it. And of course, you have all this stuff on either side, the divide for, no, whoops, that's not what I want. I wanted to copy and paste that right there. So copy, and then, no, that's giving me the wrong thing. It would have been easier, let me just rewrite it. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | And of course, you have all this stuff on either side, the divide for, no, whoops, that's not what I want. I wanted to copy and paste that right there. So copy, and then, no, that's giving me the wrong thing. It would have been easier, let me just rewrite it. That's the easiest thing. I'm having technical difficulties. So divided by 4 times 2, and on this side, you had that 7 times 2 plus this thing in orange parentheses there. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | It would have been easier, let me just rewrite it. That's the easiest thing. I'm having technical difficulties. So divided by 4 times 2, and on this side, you had that 7 times 2 plus this thing in orange parentheses there. Now, at any step, you just look again. We always want to do parentheses first. We keep wanting to do this until there's really no parentheses left. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | So divided by 4 times 2, and on this side, you had that 7 times 2 plus this thing in orange parentheses there. Now, at any step, you just look again. We always want to do parentheses first. We keep wanting to do this until there's really no parentheses left. So we have to evaluate this parentheses in orange here. So we have to evaluate this thing first. But in order to evaluate this thing, we have to look inside of it. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | We keep wanting to do this until there's really no parentheses left. So we have to evaluate this parentheses in orange here. So we have to evaluate this thing first. But in order to evaluate this thing, we have to look inside of it. And when you look inside of it, you have 7 plus 3 times 3. So if you just had 7 plus 3 times 3, how would you evaluate it? Well, look back to your order of operations. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | But in order to evaluate this thing, we have to look inside of it. And when you look inside of it, you have 7 plus 3 times 3. So if you just had 7 plus 3 times 3, how would you evaluate it? Well, look back to your order of operations. We're inside the parentheses here, so inside of it, there are no longer any parentheses. So the next thing we should do is, there are no exponents, there is multiplication. So we do that before we do any addition or subtraction. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | Well, look back to your order of operations. We're inside the parentheses here, so inside of it, there are no longer any parentheses. So the next thing we should do is, there are no exponents, there is multiplication. So we do that before we do any addition or subtraction. So we want to do the 3 times 3 before we add the 7. So this is going to be 7 plus, and the 3 times 3 we want to do first. We want to do the multiplication first. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | So we do that before we do any addition or subtraction. So we want to do the 3 times 3 before we add the 7. So this is going to be 7 plus, and the 3 times 3 we want to do first. We want to do the multiplication first. 7 plus 9. That's going to be in the orange parentheses. And then you have the 7 times 2 plus that on the left-hand side. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | We want to do the multiplication first. 7 plus 9. That's going to be in the orange parentheses. And then you have the 7 times 2 plus that on the left-hand side. You have the divided by 4 times 2 on the right-hand side. And now this, the thing in parentheses, because we still want to do the parentheses first, pretty easy to evaluate. What's 7 plus 9? |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | And then you have the 7 times 2 plus that on the left-hand side. You have the divided by 4 times 2 on the right-hand side. And now this, the thing in parentheses, because we still want to do the parentheses first, pretty easy to evaluate. What's 7 plus 9? 7 plus 9 is 16. And so everything we have simplifies to 7 times 2 plus 16 divided by 4 times 2. Now, we don't have any parentheses left, so we don't have to worry about the p in PEMDAS. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | What's 7 plus 9? 7 plus 9 is 16. And so everything we have simplifies to 7 times 2 plus 16 divided by 4 times 2. Now, we don't have any parentheses left, so we don't have to worry about the p in PEMDAS. We have no e, no exponents in this, so then we go straight to multiplication and division. We have a multiplication, we have some multiplication going on there, we have some division going on here, and a multiplication there. So we should do these next, before we do this addition right there. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | Now, we don't have any parentheses left, so we don't have to worry about the p in PEMDAS. We have no e, no exponents in this, so then we go straight to multiplication and division. We have a multiplication, we have some multiplication going on there, we have some division going on here, and a multiplication there. So we should do these next, before we do this addition right there. So we could do this multiplication, we could do that multiplication. 7 times 2 is 14. We're going to wait to do that addition. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | So we should do these next, before we do this addition right there. So we could do this multiplication, we could do that multiplication. 7 times 2 is 14. We're going to wait to do that addition. And then here we have a 16 divided by 4 times 2. That gets priority of the addition, so we're going to do that before we do the addition. But how do we evaluate that? |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | We're going to wait to do that addition. And then here we have a 16 divided by 4 times 2. That gets priority of the addition, so we're going to do that before we do the addition. But how do we evaluate that? Do we do the division first, do we do the multiplication first? And remember, I told you in the last video, when you have multiple operations of the same level, in this case division and multiplication, they're at the same level, you're safest going left to right, or you should go left to right. So you do 16 divided by 4 is 4, so this thing right here simplifies 16 divided by 4 times 2, it simplifies to 4 times 2, that's this thing in green right there. |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | But how do we evaluate that? Do we do the division first, do we do the multiplication first? And remember, I told you in the last video, when you have multiple operations of the same level, in this case division and multiplication, they're at the same level, you're safest going left to right, or you should go left to right. So you do 16 divided by 4 is 4, so this thing right here simplifies 16 divided by 4 times 2, it simplifies to 4 times 2, that's this thing in green right there. And then we're going to want to do the multiplication next. So this is going to simplify to, because multiplication takes priority of addition, this simplifies to 8, and so you get 14, this 14 right here, plus 8. And what's 14 plus 8? |
Order of operations PEMDAS Arithmetic properties Pre-Algebra Khan Academy.mp3 | So you do 16 divided by 4 is 4, so this thing right here simplifies 16 divided by 4 times 2, it simplifies to 4 times 2, that's this thing in green right there. And then we're going to want to do the multiplication next. So this is going to simplify to, because multiplication takes priority of addition, this simplifies to 8, and so you get 14, this 14 right here, plus 8. And what's 14 plus 8? That is 22. That is equal to 22. And we are done. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So let's take some things that are written in scientific notation. Just a reminder, scientific notation is useful because it allows us to write really large or really small numbers in ways that are easy for our brains to 1. write down and 2. understand. So let's write down some numbers. Let's have 3.102 times 10 to the second. I want to write it as just a numerical value. It's written in scientific notation already. It's written as a product with a power of 10. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | Let's have 3.102 times 10 to the second. I want to write it as just a numerical value. It's written in scientific notation already. It's written as a product with a power of 10. So how do I write this as just a numeral? Well, there's a slow way and a fast way. The slow way is to say, well, this is the same thing as 3.102 times 100, which means if you multiply 3.102 times 100, it'll be 3102 with two zeros behind it. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | It's written as a product with a power of 10. So how do I write this as just a numeral? Well, there's a slow way and a fast way. The slow way is to say, well, this is the same thing as 3.102 times 100, which means if you multiply 3.102 times 100, it'll be 3102 with two zeros behind it. And then we have one, two, three numbers behind the decimal point. And that would be the right answer. This is equal to 310.2. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | The slow way is to say, well, this is the same thing as 3.102 times 100, which means if you multiply 3.102 times 100, it'll be 3102 with two zeros behind it. And then we have one, two, three numbers behind the decimal point. And that would be the right answer. This is equal to 310.2. Now, a faster way to do this is just to say, well, look, right now I have only the 3 in front of the decimal point. When I take something to the times 10 to the second power, I'm essentially shifting the decimal point 2 to the right. So 3.102 times 10 to the second power is the same thing as, if I shift the decimal point 1 and then 2, because this is 10 to the second power, it's the same thing as 310.2. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | This is equal to 310.2. Now, a faster way to do this is just to say, well, look, right now I have only the 3 in front of the decimal point. When I take something to the times 10 to the second power, I'm essentially shifting the decimal point 2 to the right. So 3.102 times 10 to the second power is the same thing as, if I shift the decimal point 1 and then 2, because this is 10 to the second power, it's the same thing as 310.2. So this might be a faster way of viewing it. Every time you multiply it by 10, you shift the decimal to the right by 1. Let's do another example. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So 3.102 times 10 to the second power is the same thing as, if I shift the decimal point 1 and then 2, because this is 10 to the second power, it's the same thing as 310.2. So this might be a faster way of viewing it. Every time you multiply it by 10, you shift the decimal to the right by 1. Let's do another example. Let's say I had 7.4 times 10 to the fourth. Well, let's just do this the fast way. Let's shift the decimal 4 to the right. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | Let's do another example. Let's say I had 7.4 times 10 to the fourth. Well, let's just do this the fast way. Let's shift the decimal 4 to the right. So 7.4 times 10 to the fourth times 10 to the 1, you're going to get 74. Then times 10 to the second, you're going to get 740. We're going to have to add a 0 there because we have to shift the decimal again. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | Let's shift the decimal 4 to the right. So 7.4 times 10 to the fourth times 10 to the 1, you're going to get 74. Then times 10 to the second, you're going to get 740. We're going to have to add a 0 there because we have to shift the decimal again. 10 to the third, you're going to have 7,400. And then 10 to the fourth, you're going to have 74,000. Notice I just took this decimal and went 1, 2, 3, 4 spaces. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | We're going to have to add a 0 there because we have to shift the decimal again. 10 to the third, you're going to have 7,400. And then 10 to the fourth, you're going to have 74,000. Notice I just took this decimal and went 1, 2, 3, 4 spaces. So this is equal to 74,000. And when I had 74 and I had to shift the decimal 1 more to the right, I had to throw a 0 here. I'm multiplying it by 10. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | Notice I just took this decimal and went 1, 2, 3, 4 spaces. So this is equal to 74,000. And when I had 74 and I had to shift the decimal 1 more to the right, I had to throw a 0 here. I'm multiplying it by 10. Another way to think about it is I need 10 spaces between the leading digit and the decimal. So right here I only have 1 space. I'll need 4 spaces. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | I'm multiplying it by 10. Another way to think about it is I need 10 spaces between the leading digit and the decimal. So right here I only have 1 space. I'll need 4 spaces. So 1, 2, 3, 4. Let's do a few more examples. I think the more examples, the more you'll get what's going on. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | I'll need 4 spaces. So 1, 2, 3, 4. Let's do a few more examples. I think the more examples, the more you'll get what's going on. So I have 1.75 times 10 to the negative 3. This is in scientific notation, and I want to just write the numerical value of this. So when you take something to the negative times 10 to the negative power, you shift the decimal to the left. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | I think the more examples, the more you'll get what's going on. So I have 1.75 times 10 to the negative 3. This is in scientific notation, and I want to just write the numerical value of this. So when you take something to the negative times 10 to the negative power, you shift the decimal to the left. So this is 1.75. So if you do it times 10 to the negative 1 power, you will go 1 to the left. But if you do times 10 to the negative 2 power, you'll go 2 to the left. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So when you take something to the negative times 10 to the negative power, you shift the decimal to the left. So this is 1.75. So if you do it times 10 to the negative 1 power, you will go 1 to the left. But if you do times 10 to the negative 2 power, you'll go 2 to the left. And you'd have to put a 0 here. And if you do times 10 to the negative 3, you'd go 3 to the left. And you would have to add another 0. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | But if you do times 10 to the negative 2 power, you'll go 2 to the left. And you'd have to put a 0 here. And if you do times 10 to the negative 3, you'd go 3 to the left. And you would have to add another 0. So you take this decimal and go 1, 2, 3 to the left. So our answer would be 0.00175 is the same thing as 1.75 times 10 to the negative 3. And another way to check that you got the right answer is if you have a 1 right here, if you count the 1, 1 including the 0's to the right of the decimal should be the same as the negative exponent here. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | And you would have to add another 0. So you take this decimal and go 1, 2, 3 to the left. So our answer would be 0.00175 is the same thing as 1.75 times 10 to the negative 3. And another way to check that you got the right answer is if you have a 1 right here, if you count the 1, 1 including the 0's to the right of the decimal should be the same as the negative exponent here. So you have 1, 2, 3 numbers behind the decimal. So you should have, that's the same thing as to the negative 3 power. You're doing 1,000th. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | And another way to check that you got the right answer is if you have a 1 right here, if you count the 1, 1 including the 0's to the right of the decimal should be the same as the negative exponent here. So you have 1, 2, 3 numbers behind the decimal. So you should have, that's the same thing as to the negative 3 power. You're doing 1,000th. So this is 1,000th right there. Let's do another example. Actually, let's mix it up. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | You're doing 1,000th. So this is 1,000th right there. Let's do another example. Actually, let's mix it up. Let's start with something that's written as a numeral and then write it in scientific notation. So let's say I have 120,000. So that's just this numerical value. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | Actually, let's mix it up. Let's start with something that's written as a numeral and then write it in scientific notation. So let's say I have 120,000. So that's just this numerical value. And I want to write it in scientific notation. So this I can write as, I take the leading digit, 1.2 times 10 to the, and I just count how many digits there are behind the leading digit. 1, 2, 3, 4, 5. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So that's just this numerical value. And I want to write it in scientific notation. So this I can write as, I take the leading digit, 1.2 times 10 to the, and I just count how many digits there are behind the leading digit. 1, 2, 3, 4, 5. So 1.2 times 10 to the 5th. And if you want to kind of internalize why that makes sense, 10 to the 5th is 10,000. So 1.2, sorry, 1.2, 10 to the 5th is 100,000. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | 1, 2, 3, 4, 5. So 1.2 times 10 to the 5th. And if you want to kind of internalize why that makes sense, 10 to the 5th is 10,000. So 1.2, sorry, 1.2, 10 to the 5th is 100,000. So it's 1.2 times 1, 1, 2, 3, 4, 5. You have five 0's. That's 10 to the 5th. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So 1.2, sorry, 1.2, 10 to the 5th is 100,000. So it's 1.2 times 1, 1, 2, 3, 4, 5. You have five 0's. That's 10 to the 5th. So 1.2 times 100,000 is going to be 120,000. It's going to be 1 and 1 5th times 100,000. So 120. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | That's 10 to the 5th. So 1.2 times 100,000 is going to be 120,000. It's going to be 1 and 1 5th times 100,000. So 120. Hopefully that's sinking in. So let's do another one. Let's say the numerical value is 1,765,244. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So 120. Hopefully that's sinking in. So let's do another one. Let's say the numerical value is 1,765,244. I want to write this in scientific notation. So I take the leading digit, 1, put a decimal sign. Everything else goes behind the decimal. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | Let's say the numerical value is 1,765,244. I want to write this in scientific notation. So I take the leading digit, 1, put a decimal sign. Everything else goes behind the decimal. 7, 6, 5, 2, 4, 4. And then you count how many digits there were between the leading digit and I guess you could imagine the first decimal sign and you could have numbers that keep go over here. So between the leading digit and the decimal sign. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | Everything else goes behind the decimal. 7, 6, 5, 2, 4, 4. And then you count how many digits there were between the leading digit and I guess you could imagine the first decimal sign and you could have numbers that keep go over here. So between the leading digit and the decimal sign. And you have 1, 2, 3, 4, 5, 6 digits. So this is times 10 to the 6th. And 10 to the 6th is just a million. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So between the leading digit and the decimal sign. And you have 1, 2, 3, 4, 5, 6 digits. So this is times 10 to the 6th. And 10 to the 6th is just a million. So it's 1.765,244 times a million, which makes sense. Roughly 1.7 times a million is roughly 1.7 million. This is a little bit more than 1.7 million. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | And 10 to the 6th is just a million. So it's 1.765,244 times a million, which makes sense. Roughly 1.7 times a million is roughly 1.7 million. This is a little bit more than 1.7 million. So it makes sense. Let's do another one. How do I write 12 in scientific notation? |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | This is a little bit more than 1.7 million. So it makes sense. Let's do another one. How do I write 12 in scientific notation? Same drill. It's equal to 1.2 times, well we only have one digit between the one and the decimal spot or the decimal point. So it's 1.2 times 10 to the first power. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | How do I write 12 in scientific notation? Same drill. It's equal to 1.2 times, well we only have one digit between the one and the decimal spot or the decimal point. So it's 1.2 times 10 to the first power. Or 1.2 times 10, which is definitely equal to 12. Let's do a couple of examples where we're taking 10 to a negative power. So let's say we had 0.00281. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So it's 1.2 times 10 to the first power. Or 1.2 times 10, which is definitely equal to 12. Let's do a couple of examples where we're taking 10 to a negative power. So let's say we had 0.00281. And we want to write this in scientific notation. So what you do is you just have to think, well, how many digits are there to get to include the leading numeral in the value? So what I mean there is count 1, 2, 3. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So let's say we had 0.00281. And we want to write this in scientific notation. So what you do is you just have to think, well, how many digits are there to get to include the leading numeral in the value? So what I mean there is count 1, 2, 3. So what we want to do is we move the decimal 1, 2, 3 spaces. So one way you could think about it is you could multiply. To move the decimal to the right 3 spaces, you would multiply it by 10 to the third. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So what I mean there is count 1, 2, 3. So what we want to do is we move the decimal 1, 2, 3 spaces. So one way you could think about it is you could multiply. To move the decimal to the right 3 spaces, you would multiply it by 10 to the third. But if you're multiplying something by 10 to the third, you're changing its value. So you also have to multiply by 10 to the negative 3. Only this way will you not change the value. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | To move the decimal to the right 3 spaces, you would multiply it by 10 to the third. But if you're multiplying something by 10 to the third, you're changing its value. So you also have to multiply by 10 to the negative 3. Only this way will you not change the value. If I multiply by 10 to the 3 times 10 to the negative 3, 3 minus 3 is 0. This is just like multiplying it by 1. So what is this going to equal? |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | Only this way will you not change the value. If I multiply by 10 to the 3 times 10 to the negative 3, 3 minus 3 is 0. This is just like multiplying it by 1. So what is this going to equal? If I take the decimal and I move it 3 spaces to the right, this part right here is going to be equal to 2.81. And then we're left with this one, times 10 to the negative 3. Now, a very quick way to do it is just to say, look, let me count, including the leading numeral, how many spaces I have behind the decimal. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | So what is this going to equal? If I take the decimal and I move it 3 spaces to the right, this part right here is going to be equal to 2.81. And then we're left with this one, times 10 to the negative 3. Now, a very quick way to do it is just to say, look, let me count, including the leading numeral, how many spaces I have behind the decimal. 1, 2, 3. So it's going to be 2.81 times 10 to the negative 1, 2, 3 power. Let's do one more like that. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | Now, a very quick way to do it is just to say, look, let me count, including the leading numeral, how many spaces I have behind the decimal. 1, 2, 3. So it's going to be 2.81 times 10 to the negative 1, 2, 3 power. Let's do one more like that. Let me actually scroll up here. Let's do one more like that. Let's say I have 0 point, let's say I have 1, 2, 3, 4, 5, 6. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | Let's do one more like that. Let me actually scroll up here. Let's do one more like that. Let's say I have 0 point, let's say I have 1, 2, 3, 4, 5, 6. How many 0's do I have in this problem? Well, I'll just make up something. 0, 2, 7. |
Scientific notation examples Pre-Algebra Khan Academy.mp3 | Let's say I have 0 point, let's say I have 1, 2, 3, 4, 5, 6. How many 0's do I have in this problem? Well, I'll just make up something. 0, 2, 7. And you want to write that in scientific notation. Well, you count all the digits up to the 2 behind the decimal. So 1, 2, 3, 4, 5, 6, 7, 8. |
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