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How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | We divide both sides by the coefficient of w, multiplying w, so divide by negative 500 on the left, divide by negative 500 on the right. And we are left with w is equal to, w is equal to two. On average, women ate two bags of potato chips at these parties. We're assuming that that's constant across the parties. So let's think about how you would then figure out how many bags, on average, each man ate. Well, to do that, we just go back to either one of these equations. In the last video, or set of videos, I went to the first equation. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | We're assuming that that's constant across the parties. So let's think about how you would then figure out how many bags, on average, each man ate. Well, to do that, we just go back to either one of these equations. In the last video, or set of videos, I went to the first equation. I'll show you that the second equation should also work. Either one should work. So let's substitute back into the second equation. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | In the last video, or set of videos, I went to the first equation. I'll show you that the second equation should also work. Either one should work. So let's substitute back into the second equation. And you could either pick this version of it or this one, but I'll pick the original one. So you have 100 times m, which we're trying to figure out, plus 400 times, well, we now know that w is equal to two. 400 times two is equal to 1,100. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | So let's substitute back into the second equation. And you could either pick this version of it or this one, but I'll pick the original one. So you have 100 times m, which we're trying to figure out, plus 400 times, well, we now know that w is equal to two. 400 times two is equal to 1,100. Is equal to 1,100. So you have 100m plus 800. 800 is equal to 1,100. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | 400 times two is equal to 1,100. Is equal to 1,100. So you have 100m plus 800. 800 is equal to 1,100. And now to solve for m, we can subtract 800 from both sides. Subtract 800 from both sides. And we are left with 100m. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | 800 is equal to 1,100. And now to solve for m, we can subtract 800 from both sides. Subtract 800 from both sides. And we are left with 100m. 100m is equal to 300. And now divide both sides by 100. 100 and 100. |
How many bags of potato chips do people eat Algebra II Khan Academy.mp3 | And we are left with 100m. 100m is equal to 300. And now divide both sides by 100. 100 and 100. And we are left with m, which is, on average, the number of bags of chips each man eats is equal to three. So you have solved our bagelist's problem, what he thought was a difficult problem, using the magical, mystical powers of algebra. You are able to tell the king in his party planning process that on average, the man will eat three bags of potato chips each, and on average, the women will eat two bags of potato chips each. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | So let's just try to isolate x on one side of this inequality. But before we do that, let's just simplify this right-hand side. So we get 5x plus 7 is greater than, let's distribute this 3. So 3 times x plus 1 is the same thing as 3 times x plus 3 times 1. So it's going to be 3x plus 3 times 1 is 3. Now if we want to put our x's on the left-hand side, we can subtract 3x from both sides. That'll get rid of this 3x on the right-hand side. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | So 3 times x plus 1 is the same thing as 3 times x plus 3 times 1. So it's going to be 3x plus 3 times 1 is 3. Now if we want to put our x's on the left-hand side, we can subtract 3x from both sides. That'll get rid of this 3x on the right-hand side. So let's do that. Let's subtract 3x from both sides. And we get, on the left-hand side, 5x minus 3x is 2x plus 7 is greater than 3x minus 3x. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | That'll get rid of this 3x on the right-hand side. So let's do that. Let's subtract 3x from both sides. And we get, on the left-hand side, 5x minus 3x is 2x plus 7 is greater than 3x minus 3x. Those cancel out. That was the whole point behind subtracting 3x from both sides. Is greater than 3. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | And we get, on the left-hand side, 5x minus 3x is 2x plus 7 is greater than 3x minus 3x. Those cancel out. That was the whole point behind subtracting 3x from both sides. Is greater than 3. Now we can subtract 7 from both sides to get rid of this positive 7 right over here. So let's subtract 7 from both sides. And we get, on the left-hand side, 2x plus 7 minus 7 is just 2x. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | Is greater than 3. Now we can subtract 7 from both sides to get rid of this positive 7 right over here. So let's subtract 7 from both sides. And we get, on the left-hand side, 2x plus 7 minus 7 is just 2x. Is greater than 3 minus 7, which is negative 4. And then let's see. We have 2x is greater than negative 4. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | And we get, on the left-hand side, 2x plus 7 minus 7 is just 2x. Is greater than 3 minus 7, which is negative 4. And then let's see. We have 2x is greater than negative 4. If we just wanted x over here, we can divide both sides by 2. Since 2 is a positive number, we don't have to swap the inequality. So let's just divide both sides by 2. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | We have 2x is greater than negative 4. If we just wanted x over here, we can divide both sides by 2. Since 2 is a positive number, we don't have to swap the inequality. So let's just divide both sides by 2. And we get x is greater than negative 4 divided by 2 is negative 2. So the solution will look like this. Draw the number line. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | So let's just divide both sides by 2. And we get x is greater than negative 4 divided by 2 is negative 2. So the solution will look like this. Draw the number line. I can draw a straighter number line than that. There we go. Still not that great, but it'll serve our purposes. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | Draw the number line. I can draw a straighter number line than that. There we go. Still not that great, but it'll serve our purposes. Let's say that's negative 3, negative 2, negative 1, 0, 1, 2, 3. x is greater than negative 2. It does not include negative 2. It is not greater than or equal to negative 2. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | Still not that great, but it'll serve our purposes. Let's say that's negative 3, negative 2, negative 1, 0, 1, 2, 3. x is greater than negative 2. It does not include negative 2. It is not greater than or equal to negative 2. So we have to exclude negative 2. And we exclude negative 2 by drawing an open circle at negative 2. But then all of the values greater than that are valid x's that would satisfy this inequality. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | It is not greater than or equal to negative 2. So we have to exclude negative 2. And we exclude negative 2 by drawing an open circle at negative 2. But then all of the values greater than that are valid x's that would satisfy this inequality. So anything above it will work. And let's just try something that should work. And let's try something else that shouldn't work. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | But then all of the values greater than that are valid x's that would satisfy this inequality. So anything above it will work. And let's just try something that should work. And let's try something else that shouldn't work. So 0 should work. It is greater than negative 2. It's right over here. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | And let's try something else that shouldn't work. So 0 should work. It is greater than negative 2. It's right over here. So let's verify that. 5 times 0 plus 7 should be greater than 3 times 0 plus 1. So this is 7, because this is just a 0. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | It's right over here. So let's verify that. 5 times 0 plus 7 should be greater than 3 times 0 plus 1. So this is 7, because this is just a 0. 7 should be greater than 3. 3 times 1. So 7 should be greater than 3. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | So this is 7, because this is just a 0. 7 should be greater than 3. 3 times 1. So 7 should be greater than 3. And it definitely is. Now let's try something that should not work. Let's try negative 3. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | So 7 should be greater than 3. And it definitely is. Now let's try something that should not work. Let's try negative 3. So 5 times negative 3 plus 7. Let's see if it's greater than 3 times negative 3 plus 1. So this is negative 15 plus 7 is negative 8. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | Let's try negative 3. So 5 times negative 3 plus 7. Let's see if it's greater than 3 times negative 3 plus 1. So this is negative 15 plus 7 is negative 8. That is negative 8. Let's see if that is greater than negative 3 plus 1 is negative 2 times 3 is negative 6. Negative 8 is not greater than negative 6. |
Multi-step inequalities 2 Linear inequalities Algebra I Khan Academy.mp3 | So this is negative 15 plus 7 is negative 8. That is negative 8. Let's see if that is greater than negative 3 plus 1 is negative 2 times 3 is negative 6. Negative 8 is not greater than negative 6. Negative 8 is more negative than negative 6. It's less than. So it's good that negative 3 didn't work, because we didn't include that in our solution set. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | Let's do some exponent examples that involve division. Let's say I were to ask you what 5 to the 6th power divided by 5 to the 2nd power is. Well, we could just go to the basic definition of what an exponent represents and say, well, 5 to the 6th power, that's going to be 5 times 5 times 5 times 5 times 5. One more 5. Times 5. 5 times itself 6 times. When 5 is squared, that's just 5 times itself 2 times. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | One more 5. Times 5. 5 times itself 6 times. When 5 is squared, that's just 5 times itself 2 times. So it's going to be 5 times 5. Well, we know how to simplify a fraction or a rational expression like this. We can divide the numerator and the denominator by 1 5, and then these will cancel out. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | When 5 is squared, that's just 5 times itself 2 times. So it's going to be 5 times 5. Well, we know how to simplify a fraction or a rational expression like this. We can divide the numerator and the denominator by 1 5, and then these will cancel out. And we could do it by another 5, or this 5 and this 5 will cancel out. And what are we going to be left with? 5 times 5 times 5 times 5 over, well, you could say over 1. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | We can divide the numerator and the denominator by 1 5, and then these will cancel out. And we could do it by another 5, or this 5 and this 5 will cancel out. And what are we going to be left with? 5 times 5 times 5 times 5 over, well, you could say over 1. Or you could say that this is just 5 to the 4th power. Now notice what happens. Essentially, we started with 6 in the numerator, 6 5's multiplied by themselves in the numerator. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | 5 times 5 times 5 times 5 over, well, you could say over 1. Or you could say that this is just 5 to the 4th power. Now notice what happens. Essentially, we started with 6 in the numerator, 6 5's multiplied by themselves in the numerator. And then we subtracted out. We were able to cancel out the 2 in the denominator. So this really was equal to 5 to the 6th power minus 2. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | Essentially, we started with 6 in the numerator, 6 5's multiplied by themselves in the numerator. And then we subtracted out. We were able to cancel out the 2 in the denominator. So this really was equal to 5 to the 6th power minus 2. So we were able to subtract the exponent in the denominator from the exponent in the numerator. And let's remember how this relates to multiplication. If I had 5 to the, let me do this in different colors, 5 to the 6th times 5 to the 2nd power, we saw in the last video that this is equal to 5 to the 6 plus, I'm trying to make it color coded for you, 6 plus 2 power. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | So this really was equal to 5 to the 6th power minus 2. So we were able to subtract the exponent in the denominator from the exponent in the numerator. And let's remember how this relates to multiplication. If I had 5 to the, let me do this in different colors, 5 to the 6th times 5 to the 2nd power, we saw in the last video that this is equal to 5 to the 6 plus, I'm trying to make it color coded for you, 6 plus 2 power. Now we see a new property. And in the next video, we're going to see that these aren't really different properties, that they're kind of same sides of the same coin when we learn about negative exponents. But now in this video, we just saw that 5 to the 6th power divided by 5 to the 2nd power is going to be equal to 5 to the, it's time consuming to make it color coded for you, 6 minus 2 power. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | If I had 5 to the, let me do this in different colors, 5 to the 6th times 5 to the 2nd power, we saw in the last video that this is equal to 5 to the 6 plus, I'm trying to make it color coded for you, 6 plus 2 power. Now we see a new property. And in the next video, we're going to see that these aren't really different properties, that they're kind of same sides of the same coin when we learn about negative exponents. But now in this video, we just saw that 5 to the 6th power divided by 5 to the 2nd power is going to be equal to 5 to the, it's time consuming to make it color coded for you, 6 minus 2 power. Or 5 to the 4th power here, it's going to be 5 to the 8th. So when you multiply exponents with the same base, you add the exponents, when you divide with the same base, you subtract the denominator exponent from the numerator exponent. Let's do a bunch more of these examples right here. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | But now in this video, we just saw that 5 to the 6th power divided by 5 to the 2nd power is going to be equal to 5 to the, it's time consuming to make it color coded for you, 6 minus 2 power. Or 5 to the 4th power here, it's going to be 5 to the 8th. So when you multiply exponents with the same base, you add the exponents, when you divide with the same base, you subtract the denominator exponent from the numerator exponent. Let's do a bunch more of these examples right here. What is 6 to the 7th power divided by 6 to the 3rd power? Well once again, we can just use this property. This is going to be 6 to the 7 minus 3 power, which is equal to 6 to the 4th power. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | Let's do a bunch more of these examples right here. What is 6 to the 7th power divided by 6 to the 3rd power? Well once again, we can just use this property. This is going to be 6 to the 7 minus 3 power, which is equal to 6 to the 4th power. And you could multiply it out this way, like we did in the first problem, and verify that it indeed will be 6 to the 4th power. Now let's try something interesting. And this will be a good segue into the next video. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | This is going to be 6 to the 7 minus 3 power, which is equal to 6 to the 4th power. And you could multiply it out this way, like we did in the first problem, and verify that it indeed will be 6 to the 4th power. Now let's try something interesting. And this will be a good segue into the next video. Let's say we have 3 to the 4th power divided by 3 to the 10th power. Well if we just go from basic principles, this would be 3 times 3 times 3 times 3, all of that over 3 times 3. We're going to have 10 of these. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | And this will be a good segue into the next video. Let's say we have 3 to the 4th power divided by 3 to the 10th power. Well if we just go from basic principles, this would be 3 times 3 times 3 times 3, all of that over 3 times 3. We're going to have 10 of these. 3 times 3 times 3 times 3 times 3 times 3. How many is that? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | We're going to have 10 of these. 3 times 3 times 3 times 3 times 3 times 3. How many is that? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Well if we do what we did in the last video, this 3 cancels with that 3. Those 3's cancel, those 3's cancel, those 3's cancel. And we're left with 1 over 1, 2, 3, 4, 5, 6 3's. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Well if we do what we did in the last video, this 3 cancels with that 3. Those 3's cancel, those 3's cancel, those 3's cancel. And we're left with 1 over 1, 2, 3, 4, 5, 6 3's. So 1 over 3 to the 6th power. We have 1 over all of these 3's down here. But that property that I just told you would have told you that, look, this should also be equal to 3 to the 4 minus 10 power. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | And we're left with 1 over 1, 2, 3, 4, 5, 6 3's. So 1 over 3 to the 6th power. We have 1 over all of these 3's down here. But that property that I just told you would have told you that, look, this should also be equal to 3 to the 4 minus 10 power. Well what's 4 minus 10? We're going to get a negative number. This is 3 to the negative 6th power. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | But that property that I just told you would have told you that, look, this should also be equal to 3 to the 4 minus 10 power. Well what's 4 minus 10? We're going to get a negative number. This is 3 to the negative 6th power. So using the property we just saw, you'd get 3 to the negative 6th power. Just multiplying them out, you get 1 over 3 to the 6th power. And the fun part about all of this is these are the same quantity. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | This is 3 to the negative 6th power. So using the property we just saw, you'd get 3 to the negative 6th power. Just multiplying them out, you get 1 over 3 to the 6th power. And the fun part about all of this is these are the same quantity. So now you're learning a little bit about what it means to take a negative exponent. 3 to the negative 6th power is equal to 1 over 3 to the 6th power, and I'm going to do many, many more examples of this in the next video. But if you take anything to the negative power, so a to the negative b power is equal to 1 over a to the b. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | And the fun part about all of this is these are the same quantity. So now you're learning a little bit about what it means to take a negative exponent. 3 to the negative 6th power is equal to 1 over 3 to the 6th power, and I'm going to do many, many more examples of this in the next video. But if you take anything to the negative power, so a to the negative b power is equal to 1 over a to the b. That's one thing that we just established just now. And earlier in this video, we saw that if I have a to the b over a to the c, that this is equal to a to the b minus c. That's the other property we've been using. Now using what we've just learned and what we learned in the last video, let's do some more complicated problems. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | But if you take anything to the negative power, so a to the negative b power is equal to 1 over a to the b. That's one thing that we just established just now. And earlier in this video, we saw that if I have a to the b over a to the c, that this is equal to a to the b minus c. That's the other property we've been using. Now using what we've just learned and what we learned in the last video, let's do some more complicated problems. Let's say I have a to the third, b to the fourth power over a squared b and all of that to the third power. Well, we can use the property we just learned to simplify the inside. This is going to be equal to a to the third divided by a squared. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | Now using what we've just learned and what we learned in the last video, let's do some more complicated problems. Let's say I have a to the third, b to the fourth power over a squared b and all of that to the third power. Well, we can use the property we just learned to simplify the inside. This is going to be equal to a to the third divided by a squared. That's a to the 3 minus 2 power, right? So this would simplify to just an a. And you can imagine, this is a times a times a divided by a times a. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | This is going to be equal to a to the third divided by a squared. That's a to the 3 minus 2 power, right? So this would simplify to just an a. And you can imagine, this is a times a times a divided by a times a. You'll just have an a on top. And then the b, b to the fourth divided by b, well, that's just going to be b to the third. This is b to the first power. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | And you can imagine, this is a times a times a divided by a times a. You'll just have an a on top. And then the b, b to the fourth divided by b, well, that's just going to be b to the third. This is b to the first power. 4 minus 1 is 3. And then all of that in parentheses to the third power. I don't want to forget about this third power out here. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | This is b to the first power. 4 minus 1 is 3. And then all of that in parentheses to the third power. I don't want to forget about this third power out here. This third power is this one. Let me color code it. That third power is that one right there. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | I don't want to forget about this third power out here. This third power is this one. Let me color code it. That third power is that one right there. And then this a in orange is that a right there. And I think we understand what maps to what. And now we can use the property that when you multiply something and take the third power, that's equivalent of taking each of these. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | That third power is that one right there. And then this a in orange is that a right there. And I think we understand what maps to what. And now we can use the property that when you multiply something and take the third power, that's equivalent of taking each of these. This is equal to a to the third power times b to the third to the third power. And then this is going to be equal to a to the third power. I'll do it like this. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | And now we can use the property that when you multiply something and take the third power, that's equivalent of taking each of these. This is equal to a to the third power times b to the third to the third power. And then this is going to be equal to a to the third power. I'll do it like this. a to the third power. That's just a to the third right there. And then what is this? |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | I'll do it like this. a to the third power. That's just a to the third right there. And then what is this? Times b to the 3 times 3 power. Times b to the ninth. And we would have simplified this about as far as you can go. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | And then what is this? Times b to the 3 times 3 power. Times b to the ninth. And we would have simplified this about as far as you can go. Let's do one more of these. Because I think they're good practice and super valuable experience, I think, later on. Let's say I have 25xy to the sixth over 20y to the fifth x squared. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | And we would have simplified this about as far as you can go. Let's do one more of these. Because I think they're good practice and super valuable experience, I think, later on. Let's say I have 25xy to the sixth over 20y to the fifth x squared. So once again, we can rearrange the numerators and denominators. So this you could rewrite as 25 over 20 times x over x squared. We could have made this bottom 20x squared y to the fifth. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | Let's say I have 25xy to the sixth over 20y to the fifth x squared. So once again, we can rearrange the numerators and denominators. So this you could rewrite as 25 over 20 times x over x squared. We could have made this bottom 20x squared y to the fifth. It doesn't matter the order we do it in. Times y to the sixth over y to the fifth. And let's use our newly learned exponent properties and actually just simplifying fractions. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | We could have made this bottom 20x squared y to the fifth. It doesn't matter the order we do it in. Times y to the sixth over y to the fifth. And let's use our newly learned exponent properties and actually just simplifying fractions. 25 over 20, if you divide them both by 5, you're going to get this is equal to 5 over 4. x divided by x squared. Well, there's two ways you could think about it. That you could view as x to the negative 1. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | And let's use our newly learned exponent properties and actually just simplifying fractions. 25 over 20, if you divide them both by 5, you're going to get this is equal to 5 over 4. x divided by x squared. Well, there's two ways you could think about it. That you could view as x to the negative 1. You have a first power here. 1 minus 2 is negative 1. So this right here is equal to x to the negative 1 power. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | That you could view as x to the negative 1. You have a first power here. 1 minus 2 is negative 1. So this right here is equal to x to the negative 1 power. Or it could also be equal to 1 over x. These are equivalent. So let's say that this is equivalent to 1 over x. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | So this right here is equal to x to the negative 1 power. Or it could also be equal to 1 over x. These are equivalent. So let's say that this is equivalent to 1 over x. Just like that. And it would be x over x times x. One of those sets of x's would cancel out. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | So let's say that this is equivalent to 1 over x. Just like that. And it would be x over x times x. One of those sets of x's would cancel out. And you're just left with 1 over x. And then finally, y to the sixth over y to the fifth. That's y. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | One of those sets of x's would cancel out. And you're just left with 1 over x. And then finally, y to the sixth over y to the fifth. That's y. That right there is y to the sixth minus 5 power. Which is just y to the first power. Or just y. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | That's y. That right there is y to the sixth minus 5 power. Which is just y to the first power. Or just y. So times y. So if you want to write it all out as just one combined rational expression, you have 5 times 1 times y. Which would be 5y. |
Exponent properties involving quotients (examples) 8th grade Khan Academy.mp3 | Or just y. So times y. So if you want to write it all out as just one combined rational expression, you have 5 times 1 times y. Which would be 5y. All of that over 4 times x. This is y over 1. So 4 times x times 1. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | In this video, we're going to get introduced to the Pythagorean Theorem. Pythagorean Theorem. Which is fun on its own, but you'll see as you learn more and more mathematics, it's one of those cornerstone theorems of really all of math. It's useful in geometry, it's kind of the backbone of trigonometry. You're also going to use it to calculate distances between points. So it's a good thing to really make sure we know well. So enough talk on my end, let me tell you what the Pythagorean Theorem is. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | It's useful in geometry, it's kind of the backbone of trigonometry. You're also going to use it to calculate distances between points. So it's a good thing to really make sure we know well. So enough talk on my end, let me tell you what the Pythagorean Theorem is. So if we have a triangle, and the triangle has to be a right triangle. So it has to be a right triangle, which means that one of the three angles in the triangle have to be 90 degrees. And you specify that it's 90 degrees by drawing that little box right there. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | So enough talk on my end, let me tell you what the Pythagorean Theorem is. So if we have a triangle, and the triangle has to be a right triangle. So it has to be a right triangle, which means that one of the three angles in the triangle have to be 90 degrees. And you specify that it's 90 degrees by drawing that little box right there. So that right there is, let me do this in a different color, that right there is a 90 degree angle. Or we could call it a right angle. And a triangle that has a right angle in it is called a right triangle. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | And you specify that it's 90 degrees by drawing that little box right there. So that right there is, let me do this in a different color, that right there is a 90 degree angle. Or we could call it a right angle. And a triangle that has a right angle in it is called a right triangle. So this is called a right triangle. Now, with the Pythagorean Theorem, if we know two sides of a right triangle, we can always figure out the third side. And before I show you how to do that, let me give you one more piece of terminology. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | And a triangle that has a right angle in it is called a right triangle. So this is called a right triangle. Now, with the Pythagorean Theorem, if we know two sides of a right triangle, we can always figure out the third side. And before I show you how to do that, let me give you one more piece of terminology. The longest side of a right triangle is the side opposite the 90 degree angle, or opposite the right angle. So in this case, it is this side right here. This is the longest side. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | And before I show you how to do that, let me give you one more piece of terminology. The longest side of a right triangle is the side opposite the 90 degree angle, or opposite the right angle. So in this case, it is this side right here. This is the longest side. The way to figure out where that right triangle is, and it opens into that longest side, that longest side is called the hypotenuse. And it's good to know because we'll keep referring to it. And just so we always are good at identifying the hypotenuse, let me draw a couple of more right triangles. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | This is the longest side. The way to figure out where that right triangle is, and it opens into that longest side, that longest side is called the hypotenuse. And it's good to know because we'll keep referring to it. And just so we always are good at identifying the hypotenuse, let me draw a couple of more right triangles. So let's say I have a triangle that looks like that. A triangle that looks, let me draw it a little bit nicer. So let's say I have a triangle that looks like that. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | And just so we always are good at identifying the hypotenuse, let me draw a couple of more right triangles. So let's say I have a triangle that looks like that. A triangle that looks, let me draw it a little bit nicer. So let's say I have a triangle that looks like that. And I will tell you that this angle right here is 90 degrees. In this situation, this is the hypotenuse because it is opposite the 90 degree angle. It is the longest side. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | So let's say I have a triangle that looks like that. And I will tell you that this angle right here is 90 degrees. In this situation, this is the hypotenuse because it is opposite the 90 degree angle. It is the longest side. Let me do one more, just so that we're good at recognizing the hypotenuse. So let's say that that is my triangle. And this is the 90 degree angle right there. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | It is the longest side. Let me do one more, just so that we're good at recognizing the hypotenuse. So let's say that that is my triangle. And this is the 90 degree angle right there. And I think you know how to do this already. You go right, what it opens into, that is the hypotenuse. That is the longest side. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | And this is the 90 degree angle right there. And I think you know how to do this already. You go right, what it opens into, that is the hypotenuse. That is the longest side. So that is the hypotenuse. So once you have identified the hypotenuse, and let's say that that has length c, and now we're going to learn what the Pythagorean theorem tells us. So let's say that c is equal to the length of the hypotenuse. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | That is the longest side. So that is the hypotenuse. So once you have identified the hypotenuse, and let's say that that has length c, and now we're going to learn what the Pythagorean theorem tells us. So let's say that c is equal to the length of the hypotenuse. So let's call this c, that side is c. Let's call this side right over here a. And let's call this side over here b. So the Pythagorean theorem tells us that a squared, so one of the shorter sides, the length of one of the shorter sides squared, plus the length of the other shorter side squared, is going to be equal to the length of the hypotenuse squared. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | So let's say that c is equal to the length of the hypotenuse. So let's call this c, that side is c. Let's call this side right over here a. And let's call this side over here b. So the Pythagorean theorem tells us that a squared, so one of the shorter sides, the length of one of the shorter sides squared, plus the length of the other shorter side squared, is going to be equal to the length of the hypotenuse squared. Now, let's do that with an actual problem, and you'll see that it's actually not so bad. So let's say that I have a triangle that looks like this. Let me draw it. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | So the Pythagorean theorem tells us that a squared, so one of the shorter sides, the length of one of the shorter sides squared, plus the length of the other shorter side squared, is going to be equal to the length of the hypotenuse squared. Now, let's do that with an actual problem, and you'll see that it's actually not so bad. So let's say that I have a triangle that looks like this. Let me draw it. Let's say that this is my triangle. Looks something like this. And let's say that they tell us that this is the right angle, that this length right here, let me do this in different colors, this length right here is 3, and that this length right here is 4. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | Let me draw it. Let's say that this is my triangle. Looks something like this. And let's say that they tell us that this is the right angle, that this length right here, let me do this in different colors, this length right here is 3, and that this length right here is 4. And they want us to figure out that length right there. Now, the first thing you want to do before you even apply the Pythagorean theorem is to make sure you have your hypotenuse straight. You make sure you know what you're solving for. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | And let's say that they tell us that this is the right angle, that this length right here, let me do this in different colors, this length right here is 3, and that this length right here is 4. And they want us to figure out that length right there. Now, the first thing you want to do before you even apply the Pythagorean theorem is to make sure you have your hypotenuse straight. You make sure you know what you're solving for. And in this circumstance, we're solving for the hypotenuse. And we know that because this side over here, it is the side opposite the right angle. If we look at the Pythagorean theorem, this is c. This is the longest side. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | You make sure you know what you're solving for. And in this circumstance, we're solving for the hypotenuse. And we know that because this side over here, it is the side opposite the right angle. If we look at the Pythagorean theorem, this is c. This is the longest side. So now we're ready to apply. We're ready to apply the Pythagorean theorem. It tells us that 4 squared, one of the shorter sides, plus 3 squared, the square of another of the shorter sides, is going to be equal to this longer side squared. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | If we look at the Pythagorean theorem, this is c. This is the longest side. So now we're ready to apply. We're ready to apply the Pythagorean theorem. It tells us that 4 squared, one of the shorter sides, plus 3 squared, the square of another of the shorter sides, is going to be equal to this longer side squared. The hypotenuse squared is going to be equal to c squared. And then you just solve for c. So 4 squared is the same thing as 4 times 4. That is 16. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | It tells us that 4 squared, one of the shorter sides, plus 3 squared, the square of another of the shorter sides, is going to be equal to this longer side squared. The hypotenuse squared is going to be equal to c squared. And then you just solve for c. So 4 squared is the same thing as 4 times 4. That is 16. And 3 squared is the same thing as 3 times 3. So that is 9. And that is going to be equal to c squared. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | That is 16. And 3 squared is the same thing as 3 times 3. So that is 9. And that is going to be equal to c squared. Now, what is 16 plus 9? It's 25. So 25 is equal to c squared. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | And that is going to be equal to c squared. Now, what is 16 plus 9? It's 25. So 25 is equal to c squared. And we can take the positive square root of both sides. c, I guess, just if you look at it mathematically, could be negative 5 as well. But we're dealing with distances, so we only care about the positive roots. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | So 25 is equal to c squared. And we can take the positive square root of both sides. c, I guess, just if you look at it mathematically, could be negative 5 as well. But we're dealing with distances, so we only care about the positive roots. So you take the principal root of both sides, and you get 5 is equal to c, or the length of the longest side is equal to 5. Now, you could use the Pythagorean theorem, if we give you two of the sides to figure out the third side, no matter what the third side is. So let's do another one right over here. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | But we're dealing with distances, so we only care about the positive roots. So you take the principal root of both sides, and you get 5 is equal to c, or the length of the longest side is equal to 5. Now, you could use the Pythagorean theorem, if we give you two of the sides to figure out the third side, no matter what the third side is. So let's do another one right over here. Let's say that our triangle looks like this. And that is our right angle. Let's say that this side over here has length 12. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | So let's do another one right over here. Let's say that our triangle looks like this. And that is our right angle. Let's say that this side over here has length 12. And let's say that this side over here has length 6. And we want to figure out this length right over there. Now, like I said, the first thing you want to do is identify the hypotenuse. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | Let's say that this side over here has length 12. And let's say that this side over here has length 6. And we want to figure out this length right over there. Now, like I said, the first thing you want to do is identify the hypotenuse. And that's going to be the side opposite the right angle. We have the right angle here. You go opposite the right angle. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | Now, like I said, the first thing you want to do is identify the hypotenuse. And that's going to be the side opposite the right angle. We have the right angle here. You go opposite the right angle. The longest side, the hypotenuse, is right there. So if we think about the Pythagorean theorem, that a squared plus b squared is equal to c squared, 12 you could view as c. This is the hypotenuse. The c squared is the hypotenuse squared. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | You go opposite the right angle. The longest side, the hypotenuse, is right there. So if we think about the Pythagorean theorem, that a squared plus b squared is equal to c squared, 12 you could view as c. This is the hypotenuse. The c squared is the hypotenuse squared. So you could say 12 is equal to c. And then we could set these sides. It doesn't matter whether you call one of them a or one of them b. So let's just call this side right here. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | The c squared is the hypotenuse squared. So you could say 12 is equal to c. And then we could set these sides. It doesn't matter whether you call one of them a or one of them b. So let's just call this side right here. Let's say a is equal to 6. And then we say b, this colored b, is equal to question mark. And now we can apply the Pythagorean theorem. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | So let's just call this side right here. Let's say a is equal to 6. And then we say b, this colored b, is equal to question mark. And now we can apply the Pythagorean theorem. a squared, which is 6 squared, plus the unknown b squared, is equal to the hypotenuse squared, is equal to c squared, is equal to 12 squared. And now we can solve for b. And notice the difference here. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | And now we can apply the Pythagorean theorem. a squared, which is 6 squared, plus the unknown b squared, is equal to the hypotenuse squared, is equal to c squared, is equal to 12 squared. And now we can solve for b. And notice the difference here. Now we're not solving for the hypotenuse. We're solving for one of the shorter sides. In the last example, we solved for the hypotenuse. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | And notice the difference here. Now we're not solving for the hypotenuse. We're solving for one of the shorter sides. In the last example, we solved for the hypotenuse. We solved for c. That's why it's always important to recognize that a squared plus b squared plus c squared, c is the length of the hypotenuse. Let's just solve for b here. So we get 6 squared is 36, plus b squared is equal to 12 squared. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | In the last example, we solved for the hypotenuse. We solved for c. That's why it's always important to recognize that a squared plus b squared plus c squared, c is the length of the hypotenuse. Let's just solve for b here. So we get 6 squared is 36, plus b squared is equal to 12 squared. This 12 times 12 is 144. Now we can subtract 36 from both sides of this equation. Subtract 36. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | So we get 6 squared is 36, plus b squared is equal to 12 squared. This 12 times 12 is 144. Now we can subtract 36 from both sides of this equation. Subtract 36. Those cancel out. On the left-hand side, we're left with just a b squared is equal to 144 minus 36 is what? That is 144 minus 30 is 114. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | Subtract 36. Those cancel out. On the left-hand side, we're left with just a b squared is equal to 144 minus 36 is what? That is 144 minus 30 is 114. And then you subtract 6 is 108. So this is going to be 108. So that's what b squared is. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | That is 144 minus 30 is 114. And then you subtract 6 is 108. So this is going to be 108. So that's what b squared is. Now we want to take the principal root, or the positive root, of both sides. And you get b is equal to the square root, the principal root, of 108. Now let's see if we can simplify this a little bit. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | So that's what b squared is. Now we want to take the principal root, or the positive root, of both sides. And you get b is equal to the square root, the principal root, of 108. Now let's see if we can simplify this a little bit. The square root of 108. And what we could do is we could take the prime factorization of 108 and see how we can simplify this radical. So 108 is the same thing as 2 times 54, which is the same thing as 2 times 27, which is the same thing as 3 times 9. |
The Pythagorean theorem intro Right triangles and trigonometry Geometry Khan Academy.mp3 | Now let's see if we can simplify this a little bit. The square root of 108. And what we could do is we could take the prime factorization of 108 and see how we can simplify this radical. So 108 is the same thing as 2 times 54, which is the same thing as 2 times 27, which is the same thing as 3 times 9. So we have the square root of 108 is the same thing as the square root of 2 times 2 times, actually I'm not done, 9 can be factorized into 3 times 3. So it's 2 times 2 times 3 times 3 times 3. And so we have a couple of perfect squares in here. |
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