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So let's say that you have a to the negative four power times a to the, let's say a squared. What is that going to be? Well, once again, you have the same base. In this case, it's a. And so, and since I'm multiplying them, you can just add the exponents. So it's gonna be a to the negative four plus two power, which is equal to a to the negative two power. And once again, it should make sense. | Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 |
In this case, it's a. And so, and since I'm multiplying them, you can just add the exponents. So it's gonna be a to the negative four plus two power, which is equal to a to the negative two power. And once again, it should make sense. This right over here, that is one over a times a times a times a. And then this is times a times a. So that cancels with that, that cancels with that, and you're still left with one over a times a, which is the same thing as a to the negative two power. | Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 |
And once again, it should make sense. This right over here, that is one over a times a times a times a. And then this is times a times a. So that cancels with that, that cancels with that, and you're still left with one over a times a, which is the same thing as a to the negative two power. Now let's do it with some quotients. So what if I were to ask you, what is 12 to the negative seven divided by 12 to the negative five power? Well, when you're dividing, you subtract exponents if you have the same base. | Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 |
So that cancels with that, that cancels with that, and you're still left with one over a times a, which is the same thing as a to the negative two power. Now let's do it with some quotients. So what if I were to ask you, what is 12 to the negative seven divided by 12 to the negative five power? Well, when you're dividing, you subtract exponents if you have the same base. So this is going to be equal to 12 to the negative seven minus negative five power. You're subtracting the bottom exponent. And so this is going to be equal to 12 to the, well, subtracting a negative is the same thing as adding the positive, a 12 to the negative two power. | Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 |
Well, when you're dividing, you subtract exponents if you have the same base. So this is going to be equal to 12 to the negative seven minus negative five power. You're subtracting the bottom exponent. And so this is going to be equal to 12 to the, well, subtracting a negative is the same thing as adding the positive, a 12 to the negative two power. And once again, we just have to think about why does this actually make sense? Well, you can actually rewrite this. 12 to the negative seven divided by 12 to the negative five, that's the same thing as 12 to the negative seven times 12 to the fifth power. | Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 |
And so this is going to be equal to 12 to the, well, subtracting a negative is the same thing as adding the positive, a 12 to the negative two power. And once again, we just have to think about why does this actually make sense? Well, you can actually rewrite this. 12 to the negative seven divided by 12 to the negative five, that's the same thing as 12 to the negative seven times 12 to the fifth power. If we take the reciprocal of, if we take the reciprocal of this right over here, you would make the exponent positive. And then you get exactly what we were doing in those previous examples with products. And so let's just do one more with variables for good measure. | Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 |
12 to the negative seven divided by 12 to the negative five, that's the same thing as 12 to the negative seven times 12 to the fifth power. If we take the reciprocal of, if we take the reciprocal of this right over here, you would make the exponent positive. And then you get exactly what we were doing in those previous examples with products. And so let's just do one more with variables for good measure. Let's say I have x to the negative 20th power divided by x to the fifth power. Well, once again, we have the same base and we're taking a quotient. So this is going to be x to the negative 20 minus five because we have this one right over here in the denominator. | Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 |
And so let's just do one more with variables for good measure. Let's say I have x to the negative 20th power divided by x to the fifth power. Well, once again, we have the same base and we're taking a quotient. So this is going to be x to the negative 20 minus five because we have this one right over here in the denominator. So this is going to be equal to x to the negative 25th power. And once again, you could view our original expression as x to the negative 20th. And having an x to the fifth in the denominator, dividing by x to the fifth, is the same thing as multiplying by x to the negative five. | Multiplying & dividing powers (integer exponents) Mathematics I High School Math Khan Academy.mp3 |
He set the temperature as high as it could go. Q represents the temperature in Quinn's home in degrees Celsius after T minutes. They say Q is equal to 15 plus 0.4 T. What was the temperature when Quinn returned from vacation? So pause this video and see if you can work this out on your own. All right, so they wanna know the temperature, and you might get a little confused, say hey, maybe T is for temperature. No, T is time in minutes. Temperature is Q. Q represents the temperature. | Linear equation word problems.mp3 |
So pause this video and see if you can work this out on your own. All right, so they wanna know the temperature, and you might get a little confused, say hey, maybe T is for temperature. No, T is time in minutes. Temperature is Q. Q represents the temperature. So they really wanna know is what was Q when Quinn returned from vacation? Well, right when Quinn returned from vacation, that is when T is equal to zero. So this is equivalent to saying what is Q, our temperature, when zero minutes have elapsed? | Linear equation word problems.mp3 |
Temperature is Q. Q represents the temperature. So they really wanna know is what was Q when Quinn returned from vacation? Well, right when Quinn returned from vacation, that is when T is equal to zero. So this is equivalent to saying what is Q, our temperature, when zero minutes have elapsed? Well, if you go back to this original equation, we see that Q is equal to 15 plus 0.4 times the amount of elapsed time in minutes, so that's times zero. So that's just going to be 15 degrees Celsius. If you're familiar with slope-intercept form, you could think of it as our temperature is equal to 0.4 times the elapsed time plus 15. | Linear equation word problems.mp3 |
So this is equivalent to saying what is Q, our temperature, when zero minutes have elapsed? Well, if you go back to this original equation, we see that Q is equal to 15 plus 0.4 times the amount of elapsed time in minutes, so that's times zero. So that's just going to be 15 degrees Celsius. If you're familiar with slope-intercept form, you could think of it as our temperature is equal to 0.4 times the elapsed time plus 15. So T equals zero, you're left with just this term, which in many cases we view as our y-intercept. What is going on right when we're just getting started, right when our horizontal variable is equal to zero, and our horizontal variable in this situation is elapsed time. How much does the temperature increase every minute? | Linear equation word problems.mp3 |
If you're familiar with slope-intercept form, you could think of it as our temperature is equal to 0.4 times the elapsed time plus 15. So T equals zero, you're left with just this term, which in many cases we view as our y-intercept. What is going on right when we're just getting started, right when our horizontal variable is equal to zero, and our horizontal variable in this situation is elapsed time. How much does the temperature increase every minute? There's a couple of ways you could think about this. One, if you recognize this as slope-intercept form, you could see that 0.4 is the slope. So that says for every one minute change in time, you're going to have an increase in temperature by 0.4 degrees Celsius. | Linear equation word problems.mp3 |
How much does the temperature increase every minute? There's a couple of ways you could think about this. One, if you recognize this as slope-intercept form, you could see that 0.4 is the slope. So that says for every one minute change in time, you're going to have an increase in temperature by 0.4 degrees Celsius. So you could do it that way. You could try out some values. You could say, all right, let me think about what Q is going to be based on T. So time T equals zero, right when he got home, we already figured out that the temperature is 15 degrees Celsius. | Linear equation word problems.mp3 |
So that says for every one minute change in time, you're going to have an increase in temperature by 0.4 degrees Celsius. So you could do it that way. You could try out some values. You could say, all right, let me think about what Q is going to be based on T. So time T equals zero, right when he got home, we already figured out that the temperature is 15 degrees Celsius. At T equals one, what happens? Well, it's going to be 15 plus 0.4 times one. Well, that's just going to be 15.4. | Linear equation word problems.mp3 |
You could say, all right, let me think about what Q is going to be based on T. So time T equals zero, right when he got home, we already figured out that the temperature is 15 degrees Celsius. At T equals one, what happens? Well, it's going to be 15 plus 0.4 times one. Well, that's just going to be 15.4. Notice, when we increased our time by one, our temperature increased by 0.4 degrees Celsius, by the slope. And it would happen again. If we increased time by another minute, if we go from one to two, we would get to 15.8. | Linear equation word problems.mp3 |
Well, that's just going to be 15.4. Notice, when we increased our time by one, our temperature increased by 0.4 degrees Celsius, by the slope. And it would happen again. If we increased time by another minute, if we go from one to two, we would get to 15.8. We would increase temperature by another 0.4. How much will the temperature increase if Quinn leaves the heat on for 20 minutes? Pause the video and see if you can answer that. | Linear equation word problems.mp3 |
If we increased time by another minute, if we go from one to two, we would get to 15.8. We would increase temperature by another 0.4. How much will the temperature increase if Quinn leaves the heat on for 20 minutes? Pause the video and see if you can answer that. All right, now we have to be careful here. They're not asking us what is the temperature after 20 minutes. They're saying how much will the temperature increase if he leaves the heat on for 20 minutes. | Linear equation word problems.mp3 |
Pause the video and see if you can answer that. All right, now we have to be careful here. They're not asking us what is the temperature after 20 minutes. They're saying how much will the temperature increase if he leaves the heat on for 20 minutes. If we just want to know what is the temperature after 20 minutes, we would just say, okay, what is Q when T is equal to 20? So it'd be 15 plus 0.4 times 20. 0.4 times 20 is eight. | Linear equation word problems.mp3 |
They're saying how much will the temperature increase if he leaves the heat on for 20 minutes. If we just want to know what is the temperature after 20 minutes, we would just say, okay, what is Q when T is equal to 20? So it'd be 15 plus 0.4 times 20. 0.4 times 20 is eight. Eight plus 15 is 23. So it's 23 degrees Celsius after 20 minutes. But that's not what they're asking us. | Linear equation word problems.mp3 |
0.4 times 20 is eight. Eight plus 15 is 23. So it's 23 degrees Celsius after 20 minutes. But that's not what they're asking us. They're asking how much will the temperature increase? Well, what did we start from? We started from 15 degrees Celsius, and now after 20 minutes, we have gone to 23 degrees Celsius so we have increased by eight degrees Celsius. | Linear equation word problems.mp3 |
What I want to do in this video is think about whether the product or sums of rational numbers are definitely going to be rational. So let's just first think about the product of rational numbers. So if I have one rational number, and actually let me instead of writing out the word rational, let me just represent it as a ratio of two integers. So I have one rational number right over there. I can represent it as a over b. And I'm going to multiply it times another rational number. And I can represent that as the ratio of two integers, m and n. And so what is this product going to be? | Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 |
So I have one rational number right over there. I can represent it as a over b. And I'm going to multiply it times another rational number. And I can represent that as the ratio of two integers, m and n. And so what is this product going to be? Well, the numerator, I'm going to have am. I'm going to have a times m. And the denominator, I'm going to have b times n. b times n. Well, a is an integer, m is an integer, so you have an integer in the numerator. And b is an integer, and n is an integer, so you have an integer in the denominator. | Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 |
And I can represent that as the ratio of two integers, m and n. And so what is this product going to be? Well, the numerator, I'm going to have am. I'm going to have a times m. And the denominator, I'm going to have b times n. b times n. Well, a is an integer, m is an integer, so you have an integer in the numerator. And b is an integer, and n is an integer, so you have an integer in the denominator. So now the product is the ratio of two integers right over here. So the product is also rational. So this thing is also rational. | Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 |
And b is an integer, and n is an integer, so you have an integer in the denominator. So now the product is the ratio of two integers right over here. So the product is also rational. So this thing is also rational. So if you give me the product of any two rational numbers, you're going to end up with a rational number. Let's see if the same thing is true for the sum of two rational numbers. So let's say my first rational number is a over b, and that my second can be represented as a over b. | Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 |
So this thing is also rational. So if you give me the product of any two rational numbers, you're going to end up with a rational number. Let's see if the same thing is true for the sum of two rational numbers. So let's say my first rational number is a over b, and that my second can be represented as a over b. And my second rational number can be represented as m over n. Well, how would I add these two? Well, I can find a common denominator, and the easiest one is b times n. So let me multiply this fraction, let me multiply this one times n in the numerator and n in the denominator. And let me multiply this one times b in the numerator and b in the denominator. | Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 |
So let's say my first rational number is a over b, and that my second can be represented as a over b. And my second rational number can be represented as m over n. Well, how would I add these two? Well, I can find a common denominator, and the easiest one is b times n. So let me multiply this fraction, let me multiply this one times n in the numerator and n in the denominator. And let me multiply this one times b in the numerator and b in the denominator. Now we've written them so they have a common denominator of bn. And so this is going to be equal to an plus bm, all of that over b times n. So b times n we've just already talked about. This is definitely going to be an integer right over here. | Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 |
And let me multiply this one times b in the numerator and b in the denominator. Now we've written them so they have a common denominator of bn. And so this is going to be equal to an plus bm, all of that over b times n. So b times n we've just already talked about. This is definitely going to be an integer right over here. And then what do we have up here? Well, we have a times n, which is an integer. b times m is another integer. | Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 |
This is definitely going to be an integer right over here. And then what do we have up here? Well, we have a times n, which is an integer. b times m is another integer. The sum of two integers is going to be an integer. So you have an integer over an integer. You have the ratio of two integers. | Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 |
b times m is another integer. The sum of two integers is going to be an integer. So you have an integer over an integer. You have the ratio of two integers. So the sum of two rational numbers is going to give you another. So this one right over here was rational, and this one right over here is rational. So you take the product of two rational numbers, you get a rational number. | Sum and product of rational numbers Rational and irrational numbers Algebra I Khan Academy.mp3 |
And this is a piecewise function. It's defined as essentially different lines. You see this right over here, even with all the decimals and the negative signs, this is essentially a line. It's defined by this line over this interval for x, this line over this interval of x, and this line over this interval of x. I want to see if we can graph it. I encourage you, especially if you have some graph paper, to see if you could graph this on your own first before I work through it. So let's think about this first interval. If when negative 10 is less than or equal to x, which is less than negative two, then our function is defined by negative 0.125x plus 4.75. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
It's defined by this line over this interval for x, this line over this interval of x, and this line over this interval of x. I want to see if we can graph it. I encourage you, especially if you have some graph paper, to see if you could graph this on your own first before I work through it. So let's think about this first interval. If when negative 10 is less than or equal to x, which is less than negative two, then our function is defined by negative 0.125x plus 4.75. So this is going to be a line, a downward-sloping line. And the easiest way I can think about graphing it is let's just plot the endpoints here and then draw the line. So when x is equal to 10, so when, or sorry, when x is equal to negative 10, so we would have negative zero, actually let me write it this way. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
If when negative 10 is less than or equal to x, which is less than negative two, then our function is defined by negative 0.125x plus 4.75. So this is going to be a line, a downward-sloping line. And the easiest way I can think about graphing it is let's just plot the endpoints here and then draw the line. So when x is equal to 10, so when, or sorry, when x is equal to negative 10, so we would have negative zero, actually let me write it this way. Let me do it over here where I do the, so we're gonna have negative 0.125 times negative 10 plus 4.75. That is going to be equal to, let's see, the negative times the negative is a positive, and then 10 times this is going to be, it's going to be 1.25 plus 4.75. That is going to be equal to six. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
So when x is equal to 10, so when, or sorry, when x is equal to negative 10, so we would have negative zero, actually let me write it this way. Let me do it over here where I do the, so we're gonna have negative 0.125 times negative 10 plus 4.75. That is going to be equal to, let's see, the negative times the negative is a positive, and then 10 times this is going to be, it's going to be 1.25 plus 4.75. That is going to be equal to six. So we're going to have the point negative 10 comma six. And that point, and it includes, so x is defined there, it's less than or equal to. And then we go all the way to negative two. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
That is going to be equal to six. So we're going to have the point negative 10 comma six. And that point, and it includes, so x is defined there, it's less than or equal to. And then we go all the way to negative two. So when x is equal to negative two, we have negative 0.125 times negative two plus 4.75 is equal to, see, negative times negative is positive. Two times this is going to be, is going to be positive 0.25 plus 4.75. It's going to be equal to positive five. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
And then we go all the way to negative two. So when x is equal to negative two, we have negative 0.125 times negative two plus 4.75 is equal to, see, negative times negative is positive. Two times this is going to be, is going to be positive 0.25 plus 4.75. It's going to be equal to positive five. Now, we might be tempted, we might be tempted to just circle in this dot over here. But remember, this interval does not include negative two. It's up to it including, it's up to negative two, not including. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
It's going to be equal to positive five. Now, we might be tempted, we might be tempted to just circle in this dot over here. But remember, this interval does not include negative two. It's up to it including, it's up to negative two, not including. So I'm gonna put a little open circle there, and then I'm gonna draw the line. And then I'm gonna draw, and I'm gonna draw the line. I am going to draw my best attempt, my best attempt at the line. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
It's up to it including, it's up to negative two, not including. So I'm gonna put a little open circle there, and then I'm gonna draw the line. And then I'm gonna draw, and I'm gonna draw the line. I am going to draw my best attempt, my best attempt at the line. Now let's do the next interval. The next interval, this one's a lot more straightforward. We start at x equals negative two. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
I am going to draw my best attempt, my best attempt at the line. Now let's do the next interval. The next interval, this one's a lot more straightforward. We start at x equals negative two. When x equals negative two, negative two plus seven is, negative two plus seven is five. So negative two, so negative two comma five. So it actually includes that point right over there. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
We start at x equals negative two. When x equals negative two, negative two plus seven is, negative two plus seven is five. So negative two, so negative two comma five. So it actually includes that point right over there. So we're actually able to fill it in. And then when x is negative one, negative one plus seven is going to be positive six. Positive six, but we're not including x equals negative one, up to and including. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
So it actually includes that point right over there. So we're actually able to fill it in. And then when x is negative one, negative one plus seven is going to be positive six. Positive six, but we're not including x equals negative one, up to and including. So it's going to be, it's going to be right over here. When x is negative one, we are approaching, or as x approaches negative one, we're approaching negative one plus seven is six. So that's that interval right over there. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
Positive six, but we're not including x equals negative one, up to and including. So it's going to be, it's going to be right over here. When x is negative one, we are approaching, or as x approaches negative one, we're approaching negative one plus seven is six. So that's that interval right over there. And now let's look at this last interval. This last interval, when x is negative one, you're going to have, well this is just going to be positive 12 over 11, because we're multiplying it by negative one, plus 54 over 11, which is equal to 66 over 11, which is equal to positive six. So we're able to fill in that right over there. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
So that's that interval right over there. And now let's look at this last interval. This last interval, when x is negative one, you're going to have, well this is just going to be positive 12 over 11, because we're multiplying it by negative one, plus 54 over 11, which is equal to 66 over 11, which is equal to positive six. So we're able to fill in that right over there. And then when x is equal to 10, you have negative 120 over 11. I just multiplied this times 10. 12 times 10 is 120, and we have the negative, plus 54 over 11. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
So we're able to fill in that right over there. And then when x is equal to 10, you have negative 120 over 11. I just multiplied this times 10. 12 times 10 is 120, and we have the negative, plus 54 over 11. So this is the same thing. This is going to be, what is this? This is negative 66 over 11. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
12 times 10 is 120, and we have the negative, plus 54 over 11. So this is the same thing. This is going to be, what is this? This is negative 66 over 11. Is that right? Let's see, if you, yeah, that is negative 66 over 11, which is equal to negative six. So when x is equal to 10, our function is equal to negative negative six. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
This is negative 66 over 11. Is that right? Let's see, if you, yeah, that is negative 66 over 11, which is equal to negative six. So when x is equal to 10, our function is equal to negative negative six. And so this one actually doesn't have any jumps in it. It could've, but we see. So there we have it. | Graphing piecewise function Functions and their graphs Algebra II Khan Academy.mp3 |
So this is x is equal to negative 5. When x is equal to negative 5, y of x is equal to 6. And when x is equal to negative 2, y of x is equal to 0. So to figure out the average rate of change of y of x with respect to x, this is going to be the change in y of x over the change of x of that interval. And the shorthand for change is this triangle symbol, delta. Delta y, I'll just write y, I could write delta y of x, it's delta y, change in y over our change in x. That's going to be our average rate of change over this interval. | How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 |
So to figure out the average rate of change of y of x with respect to x, this is going to be the change in y of x over the change of x of that interval. And the shorthand for change is this triangle symbol, delta. Delta y, I'll just write y, I could write delta y of x, it's delta y, change in y over our change in x. That's going to be our average rate of change over this interval. So how much did y change over this interval? So y went from a 6 to a 0. So let's say that we can kind of view this as our end point right over here. | How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 |
That's going to be our average rate of change over this interval. So how much did y change over this interval? So y went from a 6 to a 0. So let's say that we can kind of view this as our end point right over here. So this is our end, this is our start. And we could have done it the other way around, we would get a consistent result. But since this is higher up on the list, let's call this the start, and the x is a lower value, we'll call that our start, this is our end. | How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 |
So let's say that we can kind of view this as our end point right over here. So this is our end, this is our start. And we could have done it the other way around, we would get a consistent result. But since this is higher up on the list, let's call this the start, and the x is a lower value, we'll call that our start, this is our end. So we start at 6, we end at 0. So our change in y is going to be negative 6. We went down by 6 in the y direction. | How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 |
But since this is higher up on the list, let's call this the start, and the x is a lower value, we'll call that our start, this is our end. So we start at 6, we end at 0. So our change in y is going to be negative 6. We went down by 6 in the y direction. It's negative 6, you could say that's 0 minus 6. And our change in x, well we were at negative 5 and we go up to negative 2, we increased by 3. So we increased by 3. | How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 |
We went down by 6 in the y direction. It's negative 6, you could say that's 0 minus 6. And our change in x, well we were at negative 5 and we go up to negative 2, we increased by 3. So we increased by 3. So when we increased x by 3, we decreased y of x by 6. Or if we want to simplify this right over here, negative 6 over 3 is the same thing as negative 2. So our average rate of change of y of x over the interval from negative 5 to negative 2 is negative 2. | How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 |
So we increased by 3. So when we increased x by 3, we decreased y of x by 6. Or if we want to simplify this right over here, negative 6 over 3 is the same thing as negative 2. So our average rate of change of y of x over the interval from negative 5 to negative 2 is negative 2. Every time on average x increased 1, y went down by negative 2. | How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 |
So our average rate of change of y of x over the interval from negative 5 to negative 2 is negative 2. Every time on average x increased 1, y went down by negative 2. | How to find the average rate of change from a table Functions Algebra I Khan Academy.mp3 |