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Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | So there's an infinite number of solutions. But you start to wonder, why is this happening? And so the bird whispers again into the king's ear. And the king says, well, the bird says this is because in both trips to the market, the same ratio of apples and bananas was bought. In the green trip versus the white trip, bought three times as many apples, bought three times as many bananas, and you had three times the cost. So in any situation, for any prices, per pound prices of apples and bananas, if you buy exactly three times the number of apples, three times the number of bananas, and have three times the cost, that could be true for any prices. And so this is actually consistent. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | And the king says, well, the bird says this is because in both trips to the market, the same ratio of apples and bananas was bought. In the green trip versus the white trip, bought three times as many apples, bought three times as many bananas, and you had three times the cost. So in any situation, for any prices, per pound prices of apples and bananas, if you buy exactly three times the number of apples, three times the number of bananas, and have three times the cost, that could be true for any prices. And so this is actually consistent. We can't say that our begla is lying to us. But it's not giving us enough information. This is what we call a consistent system. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | And so this is actually consistent. We can't say that our begla is lying to us. But it's not giving us enough information. This is what we call a consistent system. It's consistent information here. So let me write this down. This is consistent. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | This is what we call a consistent system. It's consistent information here. So let me write this down. This is consistent. It is consistent. 0 equals 0. There's no shadiness going on here. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | This is consistent. It is consistent. 0 equals 0. There's no shadiness going on here. But it's not enough information. This system of equations is dependent. It is dependent. |
Infinite solutions to systems Systems of equations and inequalities Algebra II Khan Academy.mp3 | There's no shadiness going on here. But it's not enough information. This system of equations is dependent. It is dependent. And you have an infinite number of solutions. Any point on this line represents a solution. So you tell our begla, well, if you really want us to figure this out, you need to give us more information. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | And they tell us to complete the table so each row represents a solution of the following equation. And they give us the equation, and then they want us to figure out what does y equal when x is equal to negative five, and what does x equal when y is equal to eight? And to figure this out, I've actually copied and pasted this part of the problem onto my scratch pad, so let me get that out. And so this is the exact same problem. And there's a couple of ways that we could try to tackle it. One way is you could try to simplify this more, get all your x's on one side and all your y's on the other side. Or we could just literally substitute when x equals negative five, what must y equal? |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | And so this is the exact same problem. And there's a couple of ways that we could try to tackle it. One way is you could try to simplify this more, get all your x's on one side and all your y's on the other side. Or we could just literally substitute when x equals negative five, what must y equal? Actually, let me do it the second way first. So if we take this equation and we substitute x with negative five, what do we get? We get negative three times, well, we're gonna say x is negative five, times negative five plus seven y is equal to five times, x is once again, it's gonna be negative five, x is negative five, five times negative five plus two y. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | Or we could just literally substitute when x equals negative five, what must y equal? Actually, let me do it the second way first. So if we take this equation and we substitute x with negative five, what do we get? We get negative three times, well, we're gonna say x is negative five, times negative five plus seven y is equal to five times, x is once again, it's gonna be negative five, x is negative five, five times negative five plus two y. See, negative three times negative five is positive 15, plus seven y is equal to negative 25 plus two y. And now to solve for y, let's see, I could subtract two y from both sides so that I get rid of the two y here on the right. So let me subtract two y, subtract two y from both sides. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | We get negative three times, well, we're gonna say x is negative five, times negative five plus seven y is equal to five times, x is once again, it's gonna be negative five, x is negative five, five times negative five plus two y. See, negative three times negative five is positive 15, plus seven y is equal to negative 25 plus two y. And now to solve for y, let's see, I could subtract two y from both sides so that I get rid of the two y here on the right. So let me subtract two y, subtract two y from both sides. And then if I want all my constants on the right-hand side, I can subtract 15 from both sides. So let me subtract 15 from both sides. And I am going to be left with 15 minus 15, that's zero. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | So let me subtract two y, subtract two y from both sides. And then if I want all my constants on the right-hand side, I can subtract 15 from both sides. So let me subtract 15 from both sides. And I am going to be left with 15 minus 15, that's zero. That's the whole point of subtracting 15 from both sides, so I get rid of this 15 here. Seven y minus two y, seven of something minus two of that same something is gonna be five of that something. It's gonna be equal to five y, is equal to negative 25 minus 15. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | And I am going to be left with 15 minus 15, that's zero. That's the whole point of subtracting 15 from both sides, so I get rid of this 15 here. Seven y minus two y, seven of something minus two of that same something is gonna be five of that something. It's gonna be equal to five y, is equal to negative 25 minus 15. Well, that's gonna be negative 40. And then two y minus two y, well, that's just gonna be zero. That was the whole point of subtracting two y from both sides. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | It's gonna be equal to five y, is equal to negative 25 minus 15. Well, that's gonna be negative 40. And then two y minus two y, well, that's just gonna be zero. That was the whole point of subtracting two y from both sides. So you have five times y is equal to negative 40, or if we divide both sides by five, we divide both sides by five, we would get y is equal to negative eight. So when x is equal to negative five, y is equal to negative eight. Y is equal to negative eight. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | That was the whole point of subtracting two y from both sides. So you have five times y is equal to negative 40, or if we divide both sides by five, we divide both sides by five, we would get y is equal to negative eight. So when x is equal to negative five, y is equal to negative eight. Y is equal to negative eight. And actually, we can fill that in. So this y is going to be equal to negative eight. And now we gotta figure this out. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | Y is equal to negative eight. And actually, we can fill that in. So this y is going to be equal to negative eight. And now we gotta figure this out. What does x equal when y is positive eight? Well, we can go back to our scratch pad here. And now let's take the same equation, but let's make y equal to positive eight. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | And now we gotta figure this out. What does x equal when y is positive eight? Well, we can go back to our scratch pad here. And now let's take the same equation, but let's make y equal to positive eight. So you have negative three x plus seven. Now y is going to be eight. Y is eight. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | And now let's take the same equation, but let's make y equal to positive eight. So you have negative three x plus seven. Now y is going to be eight. Y is eight. Seven times eight is equal to five times x plus two times, once again, y is eight. Two times eight. So we get negative three x plus 56, that's 56, is equal to five x plus 16. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | Y is eight. Seven times eight is equal to five times x plus two times, once again, y is eight. Two times eight. So we get negative three x plus 56, that's 56, is equal to five x plus 16. Now if we want to get all of our constants on one side and of all of our x terms on the other side, well, what could we do? Let's see, we could add three x to both sides. That would get rid of all of the x's on this side and put them all on this side. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | So we get negative three x plus 56, that's 56, is equal to five x plus 16. Now if we want to get all of our constants on one side and of all of our x terms on the other side, well, what could we do? Let's see, we could add three x to both sides. That would get rid of all of the x's on this side and put them all on this side. So we're gonna add three x to both sides. And let's see, if we want to get all the constants on the left-hand side, we'd want to get rid of the 16, so we could subtract 16 from the right-hand side. If we do it from the right, we're gonna have to do it from the left as well. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | That would get rid of all of the x's on this side and put them all on this side. So we're gonna add three x to both sides. And let's see, if we want to get all the constants on the left-hand side, we'd want to get rid of the 16, so we could subtract 16 from the right-hand side. If we do it from the right, we're gonna have to do it from the left as well. And we're going to be left with, these cancel out, 56 minus 16 is positive 40. And then, let's see, 16 minus 16 is zero. Five x plus three x is equal to eight x. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | If we do it from the right, we're gonna have to do it from the left as well. And we're going to be left with, these cancel out, 56 minus 16 is positive 40. And then, let's see, 16 minus 16 is zero. Five x plus three x is equal to eight x. We get eight x is equal to 40. We could divide both sides by eight. And we get five is equal to x. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | Five x plus three x is equal to eight x. We get eight x is equal to 40. We could divide both sides by eight. And we get five is equal to x. So this right over here is going to be equal to five. So let's go back. Let's go back now. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | And we get five is equal to x. So this right over here is going to be equal to five. So let's go back. Let's go back now. So when y is positive eight, x is positive five. Now they ask us, use your two solutions to graph the equation. So let's see if we can do, whoops, let me use my mouse now. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | Let's go back now. So when y is positive eight, x is positive five. Now they ask us, use your two solutions to graph the equation. So let's see if we can do, whoops, let me use my mouse now. So to graph the equation. So when x is negative five, y is negative eight. So the point negative five comma negative eight. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | So let's see if we can do, whoops, let me use my mouse now. So to graph the equation. So when x is negative five, y is negative eight. So the point negative five comma negative eight. So that's right over there. Actually, let me move my browser up so you can see that. Negative five, when x is negative five, y is negative eight. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | So the point negative five comma negative eight. So that's right over there. Actually, let me move my browser up so you can see that. Negative five, when x is negative five, y is negative eight. And when x is positive five, we see that up here, when x is positive five, y is positive eight. When x is positive five, y is positive eight. And we're done. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | Negative five, when x is negative five, y is negative eight. And when x is positive five, we see that up here, when x is positive five, y is positive eight. When x is positive five, y is positive eight. And we're done. We can check our answer if we like. We got it right. Now I said there was two ways to tackle it. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | And we're done. We can check our answer if we like. We got it right. Now I said there was two ways to tackle it. I kind of just did it, I guess you could say the naive way. I just substituted negative five directly into this and solved for y. And then I substituted y equals positive eight directly into this and then solved for x. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | Now I said there was two ways to tackle it. I kind of just did it, I guess you could say the naive way. I just substituted negative five directly into this and solved for y. And then I substituted y equals positive eight directly into this and then solved for x. Another way that I could have done it that actually probably would have been, or for sure would have been the easier way to do it, is ahead of time to try to simplify this expression. So what I could have done right from the get-go is said, hey, let's put all my x's on one side and all my y's on the other side. So this is negative three x plus seven y is equal to five x plus two y. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | And then I substituted y equals positive eight directly into this and then solved for x. Another way that I could have done it that actually probably would have been, or for sure would have been the easier way to do it, is ahead of time to try to simplify this expression. So what I could have done right from the get-go is said, hey, let's put all my x's on one side and all my y's on the other side. So this is negative three x plus seven y is equal to five x plus two y. And let's say I want to get all my y's on the left and all my x's on the right. So I don't want this negative three x on the left, so I'd want to add three x. Adding three x would cancel this out, but I can't just do it on the left-hand side, I'd have to do it on the right-hand side as well. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | So this is negative three x plus seven y is equal to five x plus two y. And let's say I want to get all my y's on the left and all my x's on the right. So I don't want this negative three x on the left, so I'd want to add three x. Adding three x would cancel this out, but I can't just do it on the left-hand side, I'd have to do it on the right-hand side as well. And then if I want to get rid of this two y on the right, I could subtract two y from the right, but of course I'd also want to do it from the left. And then what am I left with? So negative three x plus three x is zero, seven y minus two y is five y. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | Adding three x would cancel this out, but I can't just do it on the left-hand side, I'd have to do it on the right-hand side as well. And then if I want to get rid of this two y on the right, I could subtract two y from the right, but of course I'd also want to do it from the left. And then what am I left with? So negative three x plus three x is zero, seven y minus two y is five y. And then I have five x plus three x is eight x. Two y minus two y is zero. And then if I wanted to, I could solve for y, I could divide both sides by five and I'd get y is equal to 8 5ths x. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | So negative three x plus three x is zero, seven y minus two y is five y. And then I have five x plus three x is eight x. Two y minus two y is zero. And then if I wanted to, I could solve for y, I could divide both sides by five and I'd get y is equal to 8 5ths x. So this right over here represents the same exact equation as this over here, it's just written in a different way. All of the xy pairs that satisfy this would satisfy this and vice versa. And this is much easier, because if x is now negative five, if x is negative five, y would be 8 5ths times negative five. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | And then if I wanted to, I could solve for y, I could divide both sides by five and I'd get y is equal to 8 5ths x. So this right over here represents the same exact equation as this over here, it's just written in a different way. All of the xy pairs that satisfy this would satisfy this and vice versa. And this is much easier, because if x is now negative five, if x is negative five, y would be 8 5ths times negative five. Well that's going to be negative eight. And when y is equal to eight, well you actually could even do this up here. You could say five times eight is equal to eight x. |
Graphing solutions to two-variable linear equations example 2 Algebra I Khan Academy.mp3 | And this is much easier, because if x is now negative five, if x is negative five, y would be 8 5ths times negative five. Well that's going to be negative eight. And when y is equal to eight, well you actually could even do this up here. You could say five times eight is equal to eight x. And then you could see, well five times eight is the same thing as eight times five. So x would be equal to five. So I think this would actually have been a simpler way to do it. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | So let's take the number one, and let's raise it to the eighth power. So we've already seen that there's two ways of thinking about this. You could literally view this as taking eight ones and then multiplying them together. So let's do that. So you have one, two, three, four, five, six, seven, eight ones. And then you're going to multiply them together. And if you were to do that, you would get, well, 1 times 1 is 1 times 1. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | So let's do that. So you have one, two, three, four, five, six, seven, eight ones. And then you're going to multiply them together. And if you were to do that, you would get, well, 1 times 1 is 1 times 1. It doesn't matter how many times you multiply 1 by 1. You are going to just get 1. You are just going to get 1. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | And if you were to do that, you would get, well, 1 times 1 is 1 times 1. It doesn't matter how many times you multiply 1 by 1. You are going to just get 1. You are just going to get 1. And you could imagine, I did it eight times. I multiplied eight ones. But even if this was 80, or if this was 800, or if this was 8 million, if I just multiplied 1, if I had 8 million ones and I multiplied them all together, it would still be equal to 1. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | You are just going to get 1. And you could imagine, I did it eight times. I multiplied eight ones. But even if this was 80, or if this was 800, or if this was 8 million, if I just multiplied 1, if I had 8 million ones and I multiplied them all together, it would still be equal to 1. So 1 to any power is just going to be equal to 1. You would say, hey, what about 1 to the 0th power? 1 to the 0th power. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | But even if this was 80, or if this was 800, or if this was 8 million, if I just multiplied 1, if I had 8 million ones and I multiplied them all together, it would still be equal to 1. So 1 to any power is just going to be equal to 1. You would say, hey, what about 1 to the 0th power? 1 to the 0th power. Well, we've already said anything to the 0th power, except for 0. It's actually up for debate. But anything to the 0th power is going to be equal to 1. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | 1 to the 0th power. Well, we've already said anything to the 0th power, except for 0. It's actually up for debate. But anything to the 0th power is going to be equal to 1. And just as a little bit of intuition here, you could literally view this as our other definition of exponentiation, which is you start with a 1. And this number says how many times you're going to multiply that 1 times this number. So 1 times 1, 0 times, is just going to be 1. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | But anything to the 0th power is going to be equal to 1. And just as a little bit of intuition here, you could literally view this as our other definition of exponentiation, which is you start with a 1. And this number says how many times you're going to multiply that 1 times this number. So 1 times 1, 0 times, is just going to be 1. And that was a little bit clearer when we did it like this, where we said 2 to the, let's say, 4th power is equal to, this was the other definition of exponentiation we had, which is you start with a 1. And then you multiply it by 2 four times. So times 2, times 2, times 2, times 2, which is equal to, let's see, this is equal to 16. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | So 1 times 1, 0 times, is just going to be 1. And that was a little bit clearer when we did it like this, where we said 2 to the, let's say, 4th power is equal to, this was the other definition of exponentiation we had, which is you start with a 1. And then you multiply it by 2 four times. So times 2, times 2, times 2, times 2, which is equal to, let's see, this is equal to 16. So here, if you start with a 1, and then you multiply it by 1, 0 times, you're still going to have that 1 right over there. That's why anything that's not 0 to the 1 power is going to be equal to 1. Now let's try some other interesting scenarios. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | So times 2, times 2, times 2, times 2, which is equal to, let's see, this is equal to 16. So here, if you start with a 1, and then you multiply it by 1, 0 times, you're still going to have that 1 right over there. That's why anything that's not 0 to the 1 power is going to be equal to 1. Now let's try some other interesting scenarios. Let's try some negative numbers. So let's take negative 1. And let's first raise it to the 0 power. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | Now let's try some other interesting scenarios. Let's try some negative numbers. So let's take negative 1. And let's first raise it to the 0 power. So once again, this is just going based on this definition. This is starting with a 1 and then multiplying it by this number 0 times. Well, that means we're just not going to multiply it by this number. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | And let's first raise it to the 0 power. So once again, this is just going based on this definition. This is starting with a 1 and then multiplying it by this number 0 times. Well, that means we're just not going to multiply it by this number. So you're just going to get a 1. Let's try negative 1 to the 1st power. Well, anything to the 1st power, you could view this. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | Well, that means we're just not going to multiply it by this number. So you're just going to get a 1. Let's try negative 1 to the 1st power. Well, anything to the 1st power, you could view this. And I like going with this definition as opposed to this one right over here. If we were to make them consistent, if you were to make this definition consistent with this, you would say, hey, let's start with a 1. And then multiply it by 1 8 times. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | Well, anything to the 1st power, you could view this. And I like going with this definition as opposed to this one right over here. If we were to make them consistent, if you were to make this definition consistent with this, you would say, hey, let's start with a 1. And then multiply it by 1 8 times. And you're still going to get a 1 right over here. But let's do this with negative 1. So we're going to start with a 1. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | And then multiply it by 1 8 times. And you're still going to get a 1 right over here. But let's do this with negative 1. So we're going to start with a 1. And then we're going to multiply it by negative 1 1 times. Times negative 1. And this is, of course, going to be equal to negative 1. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | So we're going to start with a 1. And then we're going to multiply it by negative 1 1 times. Times negative 1. And this is, of course, going to be equal to negative 1. Now let's take negative 1. And let's take it to the 2nd power. We often say that we are squaring it when we take something to the 2nd power. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | And this is, of course, going to be equal to negative 1. Now let's take negative 1. And let's take it to the 2nd power. We often say that we are squaring it when we take something to the 2nd power. So negative 1 to the 2nd power. Well, we could start with a 1. And then multiply it by negative 1 2 times. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | We often say that we are squaring it when we take something to the 2nd power. So negative 1 to the 2nd power. Well, we could start with a 1. And then multiply it by negative 1 2 times. Multiply it by negative 1 twice. And what's this going to be equal to? And once again, by our old definition, you could also just say, hey, ignoring this 1, because that's not going to change the value. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | And then multiply it by negative 1 2 times. Multiply it by negative 1 twice. And what's this going to be equal to? And once again, by our old definition, you could also just say, hey, ignoring this 1, because that's not going to change the value. We took 2 negative 1's and we're multiplying them. Well, negative 1 times negative 1 is 1. And I think you see a pattern forming. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | And once again, by our old definition, you could also just say, hey, ignoring this 1, because that's not going to change the value. We took 2 negative 1's and we're multiplying them. Well, negative 1 times negative 1 is 1. And I think you see a pattern forming. Let's take negative 1 to the 3rd power. What's this going to be equal to? Well, by this definition, you start with a 1 and then you multiply it by negative 1 3 times. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | And I think you see a pattern forming. Let's take negative 1 to the 3rd power. What's this going to be equal to? Well, by this definition, you start with a 1 and then you multiply it by negative 1 3 times. So negative 1 times negative 1 times negative 1. Or you could just think of it as you're taking 3 negative 1's and you're multiplying it, because this 1 doesn't change the value. And this is going to be equal to negative 1 times negative 1 is positive 1 times negative 1 is negative 1. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | Well, by this definition, you start with a 1 and then you multiply it by negative 1 3 times. So negative 1 times negative 1 times negative 1. Or you could just think of it as you're taking 3 negative 1's and you're multiplying it, because this 1 doesn't change the value. And this is going to be equal to negative 1 times negative 1 is positive 1 times negative 1 is negative 1. So you see the pattern. What? Negative 1 to the 0 power is 1. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | And this is going to be equal to negative 1 times negative 1 is positive 1 times negative 1 is negative 1. So you see the pattern. What? Negative 1 to the 0 power is 1. Negative 1 of the first power is negative 1. Then you multiply by negative 1 again to get positive 1. Then you multiply it by negative 1 again to get negative 1. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | Negative 1 to the 0 power is 1. Negative 1 of the first power is negative 1. Then you multiply by negative 1 again to get positive 1. Then you multiply it by negative 1 again to get negative 1. And the pattern you might be seeing is if you take negative 1 to an odd power, you're going to get negative 1. And if you take it to an even power, you're going to get 1. you're going to get 1, because a negative times a negative is going to be the positive. And you're going to have an even number of negatives, so you're always going to have negative times negatives. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | Then you multiply it by negative 1 again to get negative 1. And the pattern you might be seeing is if you take negative 1 to an odd power, you're going to get negative 1. And if you take it to an even power, you're going to get 1. you're going to get 1, because a negative times a negative is going to be the positive. And you're going to have an even number of negatives, so you're always going to have negative times negatives. So this right over here, this is even. Even is going to be positive 1. And you could see that if you went to negative 1 to the fourth power. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | And you're going to have an even number of negatives, so you're always going to have negative times negatives. So this right over here, this is even. Even is going to be positive 1. And you could see that if you went to negative 1 to the fourth power. Negative 1 to the fourth power, well, you could start with a 1 and then multiply it by negative 1 four times. So negative 1 times negative 1 times negative 1 times negative 1, which is just going to be equal to positive 1. So if someone were to ask you, we already established that if someone were to take 1 to the 1 millionth power, this is just going to be equal to 1. |
Patterns in raising 1 and -1 to different powers Pre-Algebra Khan Academy.mp3 | And you could see that if you went to negative 1 to the fourth power. Negative 1 to the fourth power, well, you could start with a 1 and then multiply it by negative 1 four times. So negative 1 times negative 1 times negative 1 times negative 1, which is just going to be equal to positive 1. So if someone were to ask you, we already established that if someone were to take 1 to the 1 millionth power, this is just going to be equal to 1. It's just going to be equal to 1. If someone told you, let's take negative 1 and raise it to the 1 millionth power. Well, 1 million is an even number, so this is still going to be equal to positive 1. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | Let's see if we can simplify 5 times the square root of 117. So 117 doesn't jump out at me as some type of a perfect square. So let's actually take its prime factorization and see if any of those prime factors show up more than once. So it's clearly an odd number. It's clearly not divisible by 2. To test whether it's divisible by 3, we can add up all of the digits. And we explained why this works in another place on Khan Academy. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | So it's clearly an odd number. It's clearly not divisible by 2. To test whether it's divisible by 3, we can add up all of the digits. And we explained why this works in another place on Khan Academy. But if you add up all the digits, you get a 9. And 9 is divisible by 3. So 117 is going to be divisible by 3. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | And we explained why this works in another place on Khan Academy. But if you add up all the digits, you get a 9. And 9 is divisible by 3. So 117 is going to be divisible by 3. Now let's do a little aside here and figure out what 117 divided by 3 actually is. So 3 doesn't go into 1. It does go into 11 3 times. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | So 117 is going to be divisible by 3. Now let's do a little aside here and figure out what 117 divided by 3 actually is. So 3 doesn't go into 1. It does go into 11 3 times. 3 times 3 is 9. Subtract. You got a remainder of 2. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | It does go into 11 3 times. 3 times 3 is 9. Subtract. You got a remainder of 2. Bring down a 7. 3 goes into 27 9 times. 9 times 3 is 27. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | You got a remainder of 2. Bring down a 7. 3 goes into 27 9 times. 9 times 3 is 27. Subtract, and you're done. It goes in perfectly. And so we can factor 117 as 3 times 39. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | 9 times 3 is 27. Subtract, and you're done. It goes in perfectly. And so we can factor 117 as 3 times 39. Now 39 we can factor as. That jumps out more at us. That's divisible by 3. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | And so we can factor 117 as 3 times 39. Now 39 we can factor as. That jumps out more at us. That's divisible by 3. That's equivalent to 3 times 13. And then all of these are now prime numbers. So we could say that this thing is the same as 5 times the square root of 3 times 3 times 13. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | That's divisible by 3. That's equivalent to 3 times 13. And then all of these are now prime numbers. So we could say that this thing is the same as 5 times the square root of 3 times 3 times 13. And this is going to be the same thing as, and we know this from our exponent properties, as 5 times the square root of 3 times 3 times the square root of 13. Now what's the square root of 3 times 3? Well, that's the square root of 9. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | So we could say that this thing is the same as 5 times the square root of 3 times 3 times 13. And this is going to be the same thing as, and we know this from our exponent properties, as 5 times the square root of 3 times 3 times the square root of 13. Now what's the square root of 3 times 3? Well, that's the square root of 9. That's the square root of 3 squared. Any of those, well, that's just going to give you 3. So this is just going to simplify to 3. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | Well, that's the square root of 9. That's the square root of 3 squared. Any of those, well, that's just going to give you 3. So this is just going to simplify to 3. So this whole thing is 5 times 3 times the square root of 13. So this part right over here would give us 15 times the square root of 13. Let's do one more example here. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | So this is just going to simplify to 3. So this whole thing is 5 times 3 times the square root of 13. So this part right over here would give us 15 times the square root of 13. Let's do one more example here. So let's try to simplify 3 times the square root of 26. Actually, I'm going to put 26 in yellow like I did in the previous problem. 26. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | Let's do one more example here. So let's try to simplify 3 times the square root of 26. Actually, I'm going to put 26 in yellow like I did in the previous problem. 26. Well, 26 is clearly an even number. So it's going to be divisible by 2. We can rewrite it as 2 times 13. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | 26. Well, 26 is clearly an even number. So it's going to be divisible by 2. We can rewrite it as 2 times 13. And then we're done. 13 is a prime number. We can't factor this anymore. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | We can rewrite it as 2 times 13. And then we're done. 13 is a prime number. We can't factor this anymore. And so 26 doesn't have any perfect squares in it. It's not like we can factor it out as a factor of some other numbers and some perfect squares like we did here. 117 is 13 times 9. |
Simplifying square roots Exponents, radicals, and scientific notation Pre-Algebra Khan Academy.mp3 | We can't factor this anymore. And so 26 doesn't have any perfect squares in it. It's not like we can factor it out as a factor of some other numbers and some perfect squares like we did here. 117 is 13 times 9. It's the product of a perfect square and 13. 26 isn't. So we've simplified this about as much as we can. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | And you'll see that's a little bit more tricky than just the adding and subtracting numbers that we saw in the last video. And I also want to introduce you to some other types of notations for describing the solution set of an inequality. So let's do a couple of examples. So let's say I had negative 0.5 is x is less than or equal to 7.5. Now if this was an equality, your natural impulse is to say, hey, let's divide both sides by the coefficient on the x term, and that is a completely legitimate thing to do. Divide both sides by negative 0.5. The important thing you need to realize, though, when you do it with an inequality, is that when you multiply or divide both sides of the equation by a negative number, you swap the inequality. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So let's say I had negative 0.5 is x is less than or equal to 7.5. Now if this was an equality, your natural impulse is to say, hey, let's divide both sides by the coefficient on the x term, and that is a completely legitimate thing to do. Divide both sides by negative 0.5. The important thing you need to realize, though, when you do it with an inequality, is that when you multiply or divide both sides of the equation by a negative number, you swap the inequality. Think of it this way. I'll do a simple example here. If I were to tell you that 1 is less than 2, I think you would agree with that. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | The important thing you need to realize, though, when you do it with an inequality, is that when you multiply or divide both sides of the equation by a negative number, you swap the inequality. Think of it this way. I'll do a simple example here. If I were to tell you that 1 is less than 2, I think you would agree with that. 1 is definitely less than 2. Now what happens if I multiply both sides of this by negative 1? Negative 1 versus negative 2. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | If I were to tell you that 1 is less than 2, I think you would agree with that. 1 is definitely less than 2. Now what happens if I multiply both sides of this by negative 1? Negative 1 versus negative 2. Well, all of a sudden, negative 2 is more negative than negative 1. So here, negative 2 is actually less than negative 1. Now this isn't a proof, but I think it'll give you comfort on why you're swapping the sign. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | Negative 1 versus negative 2. Well, all of a sudden, negative 2 is more negative than negative 1. So here, negative 2 is actually less than negative 1. Now this isn't a proof, but I think it'll give you comfort on why you're swapping the sign. If something is larger, when you take the negative of both of it, it'll be more negative, or vice versa. So that's why if we're going to multiply both sides of this equation or divide both sides of the equation by a negative number, we need to swap the sign. So let's multiply both sides of this equation. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | Now this isn't a proof, but I think it'll give you comfort on why you're swapping the sign. If something is larger, when you take the negative of both of it, it'll be more negative, or vice versa. So that's why if we're going to multiply both sides of this equation or divide both sides of the equation by a negative number, we need to swap the sign. So let's multiply both sides of this equation. Dividing by 0.5 is the same thing as multiplying by 2. Our whole goal here is to have a 1 coefficient there. So let's multiply both sides of this equation by negative 2. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So let's multiply both sides of this equation. Dividing by 0.5 is the same thing as multiplying by 2. Our whole goal here is to have a 1 coefficient there. So let's multiply both sides of this equation by negative 2. So we have negative 2 times negative 0.5. You might say, hey, how did Sal get this 2 here? My brain is just thinking, what can I multiply negative 0.5 by to get 1? |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So let's multiply both sides of this equation by negative 2. So we have negative 2 times negative 0.5. You might say, hey, how did Sal get this 2 here? My brain is just thinking, what can I multiply negative 0.5 by to get 1? And negative 0.5 is the same thing as negative 1 half. The inverse of that is negative 2. So I'm multiplying negative 2 times both sides of this equation, and I have the 7.5 on the other side. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | My brain is just thinking, what can I multiply negative 0.5 by to get 1? And negative 0.5 is the same thing as negative 1 half. The inverse of that is negative 2. So I'm multiplying negative 2 times both sides of this equation, and I have the 7.5 on the other side. I'm going to multiply that by negative 2 as well. And remember, when you multiply or divide both sides of an inequality by a negative, you swap the inequality. You had less than or equal, now it will be greater than or equal. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So I'm multiplying negative 2 times both sides of this equation, and I have the 7.5 on the other side. I'm going to multiply that by negative 2 as well. And remember, when you multiply or divide both sides of an inequality by a negative, you swap the inequality. You had less than or equal, now it will be greater than or equal. So the left-hand side, negative 2 times negative 0.5 is just 1, you get x is greater than or equal to 7.5 times negative 2, that's negative 15, which is our solution set. All x's larger than negative 15 will satisfy this equation. I challenge you to try it. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | You had less than or equal, now it will be greater than or equal. So the left-hand side, negative 2 times negative 0.5 is just 1, you get x is greater than or equal to 7.5 times negative 2, that's negative 15, which is our solution set. All x's larger than negative 15 will satisfy this equation. I challenge you to try it. For example, 0 will work. 0 is greater than negative 15. But try something like negative 16. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | I challenge you to try it. For example, 0 will work. 0 is greater than negative 15. But try something like negative 16. Negative 16 will not work. Negative 16 times 0.5 is 8, which is not less than 7.5. So the solution set is all of the x's, let me draw a number line here, greater than negative 15. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | But try something like negative 16. Negative 16 will not work. Negative 16 times 0.5 is 8, which is not less than 7.5. So the solution set is all of the x's, let me draw a number line here, greater than negative 15. So that is negative 15 there, maybe that's negative 16, that's negative 14. Greater than or equal to negative 15 is the solution. Now, you might also see solution sets to inequalities written in interval notation. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So the solution set is all of the x's, let me draw a number line here, greater than negative 15. So that is negative 15 there, maybe that's negative 16, that's negative 14. Greater than or equal to negative 15 is the solution. Now, you might also see solution sets to inequalities written in interval notation. An interval notation, it just takes a little getting used to. We want to include negative 15, so our lower bound to our interval is negative 15. And putting this bracket here means that we're going to include negative 15. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | Now, you might also see solution sets to inequalities written in interval notation. An interval notation, it just takes a little getting used to. We want to include negative 15, so our lower bound to our interval is negative 15. And putting this bracket here means that we're going to include negative 15. The set includes the bottom boundary, it includes negative 15. And we're going to go all the way to infinity. And we put a curly, a parentheses here. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | And putting this bracket here means that we're going to include negative 15. The set includes the bottom boundary, it includes negative 15. And we're going to go all the way to infinity. And we put a curly, a parentheses here. Parentheses normally means that you're not including the upper bound. You also do it for infinity, because infinity really isn't a normal number, so to speak. You can't just say, oh, I'm at infinity. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | And we put a curly, a parentheses here. Parentheses normally means that you're not including the upper bound. You also do it for infinity, because infinity really isn't a normal number, so to speak. You can't just say, oh, I'm at infinity. You're never at infinity. So that's why you put that parentheses. But the parentheses tends to mean that you don't include that boundary, but you also use it with infinity. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | You can't just say, oh, I'm at infinity. You're never at infinity. So that's why you put that parentheses. But the parentheses tends to mean that you don't include that boundary, but you also use it with infinity. So this and this are the exact same thing. Sometimes you might also see set notations, where the solution of that, they might say x is a real number such that, that little line, that vertical line thing, just means such that x is greater than or equal to negative 15. So this is the set, these curly brackets mean the set of all real numbers, or the set of all numbers where x is a real number such that x is greater than or equal to negative 15. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | But the parentheses tends to mean that you don't include that boundary, but you also use it with infinity. So this and this are the exact same thing. Sometimes you might also see set notations, where the solution of that, they might say x is a real number such that, that little line, that vertical line thing, just means such that x is greater than or equal to negative 15. So this is the set, these curly brackets mean the set of all real numbers, or the set of all numbers where x is a real number such that x is greater than or equal to negative 15. All of this, this, and this are all equivalent. Let's keep that in mind and do a couple of more examples. So let's say we had 75x is greater than or equal to 125. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So this is the set, these curly brackets mean the set of all real numbers, or the set of all numbers where x is a real number such that x is greater than or equal to negative 15. All of this, this, and this are all equivalent. Let's keep that in mind and do a couple of more examples. So let's say we had 75x is greater than or equal to 125. So here we can just divide both sides by 75. And since 75 is a positive number, you don't have to change the inequality. So you get x is greater than or equal to 125 over 75. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So let's say we had 75x is greater than or equal to 125. So here we can just divide both sides by 75. And since 75 is a positive number, you don't have to change the inequality. So you get x is greater than or equal to 125 over 75. And if you divide the numerator and denominator by 25, this is 5 over 3. So x is greater than or equal to 5 thirds. Or we could write the solution set being from including 5 thirds to infinity. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | So you get x is greater than or equal to 125 over 75. And if you divide the numerator and denominator by 25, this is 5 over 3. So x is greater than or equal to 5 thirds. Or we could write the solution set being from including 5 thirds to infinity. And once again, if you were to graph it on a number line, 5 thirds is what? That's 1 and 2 thirds. So you have 0, 1, 2, and 1 and 2 thirds will be right around there. |
Multiplying and dividing with inequalities Linear inequalities Algebra I Khan Academy.mp3 | Or we could write the solution set being from including 5 thirds to infinity. And once again, if you were to graph it on a number line, 5 thirds is what? That's 1 and 2 thirds. So you have 0, 1, 2, and 1 and 2 thirds will be right around there. We're going to include it. That right there is 5 thirds. And everything greater than or equal to that will be included in our solution set. |