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Example two-step equation with numerator x Linear equations Algebra I Khan Academy.mp3 | I could write just a plus 0, but I think that's a little unnecessary. And so we have negative 18 is equal to x over 4, and our whole goal here is to isolate the x, to solve for the x. And the best way we could do that, if we have x over 4 here, if we multiply that by 4, we're just going to have an x. So we can multiply that by 4, but once again, this is an equation. Anything you do to the right-hand side, you have to do to the left-hand side, and vice versa. So if we multiply the right-hand side by 4, we also have to multiply the left-hand side by 4. So we get 4 times negative 18 is equal to x over 4 times 4. |
Example two-step equation with numerator x Linear equations Algebra I Khan Academy.mp3 | So we can multiply that by 4, but once again, this is an equation. Anything you do to the right-hand side, you have to do to the left-hand side, and vice versa. So if we multiply the right-hand side by 4, we also have to multiply the left-hand side by 4. So we get 4 times negative 18 is equal to x over 4 times 4. The x over 4 times 4, that cancels out. You divide something by 4 and multiply by 4. You're just going to be left with an x. |
Example two-step equation with numerator x Linear equations Algebra I Khan Academy.mp3 | So we get 4 times negative 18 is equal to x over 4 times 4. The x over 4 times 4, that cancels out. You divide something by 4 and multiply by 4. You're just going to be left with an x. And on the other side, 4 times negative 18, let's see, that's 40. Well, let's just write it out. So 18 times 4, if we were to multiply 18 times 4, 4 times 8 is 32. |
Example two-step equation with numerator x Linear equations Algebra I Khan Academy.mp3 | You're just going to be left with an x. And on the other side, 4 times negative 18, let's see, that's 40. Well, let's just write it out. So 18 times 4, if we were to multiply 18 times 4, 4 times 8 is 32. 4 times 1 is 4, plus 1 is 72. But this is negative 18 times 4, so it's negative 72. So x is equal to negative 72. |
Example two-step equation with numerator x Linear equations Algebra I Khan Academy.mp3 | So 18 times 4, if we were to multiply 18 times 4, 4 times 8 is 32. 4 times 1 is 4, plus 1 is 72. But this is negative 18 times 4, so it's negative 72. So x is equal to negative 72. And if we want to check it, we can just substitute it back into that original equation. So let's do that. Let's substitute this into the original equation. |
Example two-step equation with numerator x Linear equations Algebra I Khan Academy.mp3 | So x is equal to negative 72. And if we want to check it, we can just substitute it back into that original equation. So let's do that. Let's substitute this into the original equation. So the original equation was negative 16 is equal to, instead of writing x, I'm going to write negative 72, is equal to negative 72 over 4 plus 2. Let's see if this is actually true. So this right-hand side simplifies to negative 72 divided by 4. |
Example two-step equation with numerator x Linear equations Algebra I Khan Academy.mp3 | Let's substitute this into the original equation. So the original equation was negative 16 is equal to, instead of writing x, I'm going to write negative 72, is equal to negative 72 over 4 plus 2. Let's see if this is actually true. So this right-hand side simplifies to negative 72 divided by 4. We already know that that is negative 18. So this is equal to negative 18 plus 2. This is what the equation becomes. |
Example two-step equation with numerator x Linear equations Algebra I Khan Academy.mp3 | So this right-hand side simplifies to negative 72 divided by 4. We already know that that is negative 18. So this is equal to negative 18 plus 2. This is what the equation becomes. And then the right-hand side, negative 18 plus 2, that's negative 16. So it all comes out true. This right-hand side, when x is equal to negative 72, does indeed equal negative 16. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | So any line can be represented in slope intercept form as y is equal to mx plus b. Where this m right over here, that is the slope of the line. And this b over here, this is the y intercept of the line. Let me draw a quick line here just so that we can visualize that a little bit. So that is my y axis and then that is my x axis. Let me draw a line. Since our line here has a negative slope, I'll draw a downward sloping line. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | Let me draw a quick line here just so that we can visualize that a little bit. So that is my y axis and then that is my x axis. Let me draw a line. Since our line here has a negative slope, I'll draw a downward sloping line. Let's say our line looks something like that. Hopefully we're a little familiar with the slope already. The slope essentially tells us start at some point on the line and go to some other point on the line. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | Since our line here has a negative slope, I'll draw a downward sloping line. Let's say our line looks something like that. Hopefully we're a little familiar with the slope already. The slope essentially tells us start at some point on the line and go to some other point on the line. Measure how much you have to move in the x direction, that is your run. And then measure how much you have to move in the y direction, that is your rise. And our slope is equal to rise over run. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | The slope essentially tells us start at some point on the line and go to some other point on the line. Measure how much you have to move in the x direction, that is your run. And then measure how much you have to move in the y direction, that is your rise. And our slope is equal to rise over run. What you're going to see over here would be downward sloping. Because if you move in the positive x direction, we have to go down. If our run is positive, our rise here is negative. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | And our slope is equal to rise over run. What you're going to see over here would be downward sloping. Because if you move in the positive x direction, we have to go down. If our run is positive, our rise here is negative. So this would be a negative over positive, give you a negative number. That makes sense because we're downward sloping. The more we go down in this situation, for every step we move to the right, the more downward sloping we'll be, the more of a negative slope we'll have. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | If our run is positive, our rise here is negative. So this would be a negative over positive, give you a negative number. That makes sense because we're downward sloping. The more we go down in this situation, for every step we move to the right, the more downward sloping we'll be, the more of a negative slope we'll have. That's slope. The y intercept just tells us where we intercept the y axis. The y intercept, this point right over here, this is where the line intersects with the y axis. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | The more we go down in this situation, for every step we move to the right, the more downward sloping we'll be, the more of a negative slope we'll have. That's slope. The y intercept just tells us where we intercept the y axis. The y intercept, this point right over here, this is where the line intersects with the y axis. This will be the point 0, b. This actually just falls straight out of this equation. When x is equal to 0, let's evaluate this equation when x is equal to 0. y will be equal to m times 0 plus b. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | The y intercept, this point right over here, this is where the line intersects with the y axis. This will be the point 0, b. This actually just falls straight out of this equation. When x is equal to 0, let's evaluate this equation when x is equal to 0. y will be equal to m times 0 plus b. Anything times 0 is 0. y is equal to 0 plus b, or y will be equal to b when x is equal to 0. When x is equal to 0. This is the point 0, b. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | When x is equal to 0, let's evaluate this equation when x is equal to 0. y will be equal to m times 0 plus b. Anything times 0 is 0. y is equal to 0 plus b, or y will be equal to b when x is equal to 0. When x is equal to 0. This is the point 0, b. They tell us what the slope of this line is. They tell us a line has a slope of negative 3 4ths. We know that our slope is negative 3 4ths. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | This is the point 0, b. They tell us what the slope of this line is. They tell us a line has a slope of negative 3 4ths. We know that our slope is negative 3 4ths. They tell us that the line goes through the point 0, 8. They tell us we go through the point 0, 8. Notice x is 0. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | We know that our slope is negative 3 4ths. They tell us that the line goes through the point 0, 8. They tell us we go through the point 0, 8. Notice x is 0. We're on the y axis. When x is 0, we're on the y axis. This is our y intercept. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | Notice x is 0. We're on the y axis. When x is 0, we're on the y axis. This is our y intercept. Our y intercept is the point 0, 8. We could say that b is equal to 8. We know m is equal to negative 3 4ths. |
Slope-intercept equation from slope and point Algebra I Khan Academy.mp3 | This is our y intercept. Our y intercept is the point 0, 8. We could say that b is equal to 8. We know m is equal to negative 3 4ths. b is equal to 8. We can write the equation of this line in slope intercept form. y is equal to negative 3 4ths times x plus b, plus 8. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | But before I even do that, let's have a little bit of a review of what an exponent even is. So let's say I had 2 to the third power. You might be tempted to say, oh, is that 6? And I would say, no, it is not 6. This means 2 times itself 3 times. So this is going to be equal to 2 times 2 times 2, which is equal to 2 times 2 is 4. 4 times 2 is equal to 8. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | And I would say, no, it is not 6. This means 2 times itself 3 times. So this is going to be equal to 2 times 2 times 2, which is equal to 2 times 2 is 4. 4 times 2 is equal to 8. If I were to ask you what 3 to the second power is, or 3 squared, this is equal to 3 times itself 2 times. This is equal to 3 times 3, which is equal to 9. Let's do one more of these. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | 4 times 2 is equal to 8. If I were to ask you what 3 to the second power is, or 3 squared, this is equal to 3 times itself 2 times. This is equal to 3 times 3, which is equal to 9. Let's do one more of these. I think you're getting the general sense if you've never seen these before. Let's say I have 5 to the seventh power. That's equal to 5 times itself 7 times. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Let's do one more of these. I think you're getting the general sense if you've never seen these before. Let's say I have 5 to the seventh power. That's equal to 5 times itself 7 times. 5 times 5 times 5 times 5 times 5 times 5 times 5. That's 7, right? 1, 2, 3, 4, 5, 6, 7. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | That's equal to 5 times itself 7 times. 5 times 5 times 5 times 5 times 5 times 5 times 5. That's 7, right? 1, 2, 3, 4, 5, 6, 7. This is going to be a really, really, really large number. I'm not going to calculate it right now. If you want to do it by hand, feel free to do so or use a calculator. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | 1, 2, 3, 4, 5, 6, 7. This is going to be a really, really, really large number. I'm not going to calculate it right now. If you want to do it by hand, feel free to do so or use a calculator. But this is a really, really, really large number. One thing that you might appreciate very quickly is that exponents increase very rapidly. 5 to the 17th would be even a way, way more massive number. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | If you want to do it by hand, feel free to do so or use a calculator. But this is a really, really, really large number. One thing that you might appreciate very quickly is that exponents increase very rapidly. 5 to the 17th would be even a way, way more massive number. But anyway, that's a review of exponents. Let's get a little bit steeped in algebra using exponents. What would 3x times 3x times 3x be? |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | 5 to the 17th would be even a way, way more massive number. But anyway, that's a review of exponents. Let's get a little bit steeped in algebra using exponents. What would 3x times 3x times 3x be? One thing you need to remember about multiplication is it doesn't matter what order you do the multiplication in. This is going to be the same thing as 3 times 3 times 3 times x times x times x. Based on what we reviewed just here, that part right there, 3 times 3, 3 times, that's 3 to the 3rd power. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | What would 3x times 3x times 3x be? One thing you need to remember about multiplication is it doesn't matter what order you do the multiplication in. This is going to be the same thing as 3 times 3 times 3 times x times x times x. Based on what we reviewed just here, that part right there, 3 times 3, 3 times, that's 3 to the 3rd power. This right here, x times itself 3 times, that's x to the 3rd power. This whole thing can be written as 3 to the 3rd times x to the 3rd. Or if you know what 3 to the 3rd is, this is 9 times 3, which is 27. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Based on what we reviewed just here, that part right there, 3 times 3, 3 times, that's 3 to the 3rd power. This right here, x times itself 3 times, that's x to the 3rd power. This whole thing can be written as 3 to the 3rd times x to the 3rd. Or if you know what 3 to the 3rd is, this is 9 times 3, which is 27. This is 27x to the 3rd power. Now you might have said, hey, wasn't 3x times 3x times 3x, wasn't that 3x to the 3rd power? You're multiplying 3x times itself 3 times. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Or if you know what 3 to the 3rd is, this is 9 times 3, which is 27. This is 27x to the 3rd power. Now you might have said, hey, wasn't 3x times 3x times 3x, wasn't that 3x to the 3rd power? You're multiplying 3x times itself 3 times. And I would say, yes it is. So this right here, you could interpret that as 3x to the 3rd power. And just like that, we stumbled on one of our exponent properties. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | You're multiplying 3x times itself 3 times. And I would say, yes it is. So this right here, you could interpret that as 3x to the 3rd power. And just like that, we stumbled on one of our exponent properties. Notice this, when I have something times something and that whole thing is to the 3rd power, that equals each of those things to the 3rd power times each other. So 3x to the 3rd is the same thing as 3 to the 3rd times x to the 3rd, which is 27 to the 3rd power. Let's do a couple more examples. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | And just like that, we stumbled on one of our exponent properties. Notice this, when I have something times something and that whole thing is to the 3rd power, that equals each of those things to the 3rd power times each other. So 3x to the 3rd is the same thing as 3 to the 3rd times x to the 3rd, which is 27 to the 3rd power. Let's do a couple more examples. What if I were to ask you what 6 to the 3rd times 6 to the 6th power is? This is going to be a really huge number, but I want to write it as a power of 6. Let me write the 6 to the 6th actually in a different color. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Let's do a couple more examples. What if I were to ask you what 6 to the 3rd times 6 to the 6th power is? This is going to be a really huge number, but I want to write it as a power of 6. Let me write the 6 to the 6th actually in a different color. 6 to the 3rd times 6 to the 6th power. What is this going to be equal to? 6 to the 3rd, we know that's 6 times itself 3 times. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Let me write the 6 to the 6th actually in a different color. 6 to the 3rd times 6 to the 6th power. What is this going to be equal to? 6 to the 3rd, we know that's 6 times itself 3 times. So it's 6 times 6 times 6. And then that's going to be times, the times here is in green, so I'll do it in green. Maybe I'll make both of them in orange. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | 6 to the 3rd, we know that's 6 times itself 3 times. So it's 6 times 6 times 6. And then that's going to be times, the times here is in green, so I'll do it in green. Maybe I'll make both of them in orange. That is going to be times 6 to the 6th power. What's 6 to the 6th power? That's 6 times itself 6 times. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Maybe I'll make both of them in orange. That is going to be times 6 to the 6th power. What's 6 to the 6th power? That's 6 times itself 6 times. So it's 6 times 6 times 6 times 6 times 6. Then you get one more times 6. What is this whole number going to be? |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | That's 6 times itself 6 times. So it's 6 times 6 times 6 times 6 times 6. Then you get one more times 6. What is this whole number going to be? This whole thing, we're multiplying 6 times itself how many times? 1, 2, 3, 4, 5, 6, 7, 8, 9 times. 3 times here and then another 6 times here. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | What is this whole number going to be? This whole thing, we're multiplying 6 times itself how many times? 1, 2, 3, 4, 5, 6, 7, 8, 9 times. 3 times here and then another 6 times here. We're multiplying 6 times itself 9 times. 3 plus 6. So this is equal to 6 to the 3 plus 6th power or 6 to the 9th power. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | 3 times here and then another 6 times here. We're multiplying 6 times itself 9 times. 3 plus 6. So this is equal to 6 to the 3 plus 6th power or 6 to the 9th power. And just like that, we've stumbled on another exponent property. When we take exponents, in this case 6 to the 3rd, the number 6 is the base. We're taking the base to the exponent of 3. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | So this is equal to 6 to the 3 plus 6th power or 6 to the 9th power. And just like that, we've stumbled on another exponent property. When we take exponents, in this case 6 to the 3rd, the number 6 is the base. We're taking the base to the exponent of 3. When you have the same base and you're multiplying 2 exponents with the same base, you can add the exponents. If I have, let me do several more examples of this. If I have, let's do it in magenta. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | We're taking the base to the exponent of 3. When you have the same base and you're multiplying 2 exponents with the same base, you can add the exponents. If I have, let me do several more examples of this. If I have, let's do it in magenta. Let's say I had 2 squared times 2 to the 4th times 2 to the 6th. I have the same base in all of these, so I can add the exponents. This is going to be equal to 2 to the 2 plus 4 plus 6, which is equal to 2 to the 12th power. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | If I have, let's do it in magenta. Let's say I had 2 squared times 2 to the 4th times 2 to the 6th. I have the same base in all of these, so I can add the exponents. This is going to be equal to 2 to the 2 plus 4 plus 6, which is equal to 2 to the 12th power. Hopefully that makes sense because this is going to be 2 times itself 2 times, 2 times itself 4 times, 2 times itself 6 times. When you multiply them all out, it's going to be 2 times itself 12 times or 2 to the 12th power. Let's do it in a little bit more abstract way using some variables, but it's the same exact idea. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | This is going to be equal to 2 to the 2 plus 4 plus 6, which is equal to 2 to the 12th power. Hopefully that makes sense because this is going to be 2 times itself 2 times, 2 times itself 4 times, 2 times itself 6 times. When you multiply them all out, it's going to be 2 times itself 12 times or 2 to the 12th power. Let's do it in a little bit more abstract way using some variables, but it's the same exact idea. What is x to the squared or x squared times x to the 4th? We could use the property we just learned. We have the exact same base, x. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Let's do it in a little bit more abstract way using some variables, but it's the same exact idea. What is x to the squared or x squared times x to the 4th? We could use the property we just learned. We have the exact same base, x. It's going to be x to the 2 plus 4 power. It's going to be x to the 6th power. If you don't believe me, what is x squared? |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | We have the exact same base, x. It's going to be x to the 2 plus 4 power. It's going to be x to the 6th power. If you don't believe me, what is x squared? x squared is equal to x times x. If you're going to multiply that times x to the 4th, you're multiplying it by x times itself 4 times, x times x times x times x. How many times are you now multiplying x by itself? |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | If you don't believe me, what is x squared? x squared is equal to x times x. If you're going to multiply that times x to the 4th, you're multiplying it by x times itself 4 times, x times x times x times x. How many times are you now multiplying x by itself? 1, 2, 3, 4, 5, 6 times x to the 6th power. Let's do another one of these. The more examples you see, I figure the better. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | How many times are you now multiplying x by itself? 1, 2, 3, 4, 5, 6 times x to the 6th power. Let's do another one of these. The more examples you see, I figure the better. Let me do the other property just to mix and match it. Let's say I have a to the 3rd to the 4th power. I'll tell you the property here, and then I'll show you why it makes sense. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | The more examples you see, I figure the better. Let me do the other property just to mix and match it. Let's say I have a to the 3rd to the 4th power. I'll tell you the property here, and then I'll show you why it makes sense. When you have something to an exponent, and then you raise that to an exponent, you can multiply the exponent. This is going to be a to the 3 times 4 power, or a to the 12th power. Why does that make sense? |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | I'll tell you the property here, and then I'll show you why it makes sense. When you have something to an exponent, and then you raise that to an exponent, you can multiply the exponent. This is going to be a to the 3 times 4 power, or a to the 12th power. Why does that make sense? This right here is a to the 3rd times itself 4 times. This is equal to a to the 3rd times a to the 3rd times a to the 3rd times a to the 3rd. We have the same base, so we can add the exponents. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Why does that make sense? This right here is a to the 3rd times itself 4 times. This is equal to a to the 3rd times a to the 3rd times a to the 3rd times a to the 3rd. We have the same base, so we can add the exponents. This is going to be a to the 3 times 4. This is equal to a to the 3 plus 3 plus 3 plus 3 power, which is the same thing as a to the 3 times 4 power, or a to the 12th power. Just to review the properties we've learned so far in this video, besides just a review of what an exponent is, if I have something, let's say I have x to the a power times x to the b power, this is going to be equal to x to the a plus b power. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | We have the same base, so we can add the exponents. This is going to be a to the 3 times 4. This is equal to a to the 3 plus 3 plus 3 plus 3 power, which is the same thing as a to the 3 times 4 power, or a to the 12th power. Just to review the properties we've learned so far in this video, besides just a review of what an exponent is, if I have something, let's say I have x to the a power times x to the b power, this is going to be equal to x to the a plus b power. We saw that right here. x squared times x to the 4th is equal to x to the 6, 2 plus 4. We also saw that if I have x times y to the a power, this is the same thing as x to the a power times y to the a power. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Just to review the properties we've learned so far in this video, besides just a review of what an exponent is, if I have something, let's say I have x to the a power times x to the b power, this is going to be equal to x to the a plus b power. We saw that right here. x squared times x to the 4th is equal to x to the 6, 2 plus 4. We also saw that if I have x times y to the a power, this is the same thing as x to the a power times y to the a power. We saw that early on in this video. We saw that over here. 3x to the 3rd is the same thing as 3 to the 3rd times x to the 3rd. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | We also saw that if I have x times y to the a power, this is the same thing as x to the a power times y to the a power. We saw that early on in this video. We saw that over here. 3x to the 3rd is the same thing as 3 to the 3rd times x to the 3rd. That's what this is saying right here. 3x to the 3rd is the same thing as 3 to the 3rd times x to the 3rd. The last property, which we just stumbled upon, is if you have x to the a, and then you raise that to the b power, that's equal to x to the a times b. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | 3x to the 3rd is the same thing as 3 to the 3rd times x to the 3rd. That's what this is saying right here. 3x to the 3rd is the same thing as 3 to the 3rd times x to the 3rd. The last property, which we just stumbled upon, is if you have x to the a, and then you raise that to the b power, that's equal to x to the a times b. We saw that right there. a to the 3rd, and then raise that to the 4th power, is the same thing as a to the 3 times 4, or a to the 12th power. Let's use these properties to do a handful of what we could call more complex problems. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | The last property, which we just stumbled upon, is if you have x to the a, and then you raise that to the b power, that's equal to x to the a times b. We saw that right there. a to the 3rd, and then raise that to the 4th power, is the same thing as a to the 3 times 4, or a to the 12th power. Let's use these properties to do a handful of what we could call more complex problems. Let's say we had 2xy squared times negative x squared y squared times 3x squared y squared. We wanted to simplify this. A good place to start, maybe we could simplify this. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Let's use these properties to do a handful of what we could call more complex problems. Let's say we had 2xy squared times negative x squared y squared times 3x squared y squared. We wanted to simplify this. A good place to start, maybe we could simplify this. This you could view as negative 1 times x squared times y squared. Just this part right here, if we take this whole thing to the squared power, this is like raising each of these to the second power. This part right here could be simplified as negative 1 squared times x squared squared times y squared. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | A good place to start, maybe we could simplify this. This you could view as negative 1 times x squared times y squared. Just this part right here, if we take this whole thing to the squared power, this is like raising each of these to the second power. This part right here could be simplified as negative 1 squared times x squared squared times y squared. If we were to simplify that, negative 1 squared is just 1. x squared squared, remember, you can just multiply the exponents. That's going to be x to the 4th y squared. That's what this middle part simplifies to. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | This part right here could be simplified as negative 1 squared times x squared squared times y squared. If we were to simplify that, negative 1 squared is just 1. x squared squared, remember, you can just multiply the exponents. That's going to be x to the 4th y squared. That's what this middle part simplifies to. Let's see if we can merge it with the other parts. The other parts to remember were 2xy squared and then 3x squared y squared. Now we're going ahead and just straight up multiplying everything. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | That's what this middle part simplifies to. Let's see if we can merge it with the other parts. The other parts to remember were 2xy squared and then 3x squared y squared. Now we're going ahead and just straight up multiplying everything. We learned in multiplication that it doesn't matter which order you multiply things in. I can just rearrange. We're just going and multiplying 2 times x times y squared times x to the 4th times y squared times 3 times x squared times y squared. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Now we're going ahead and just straight up multiplying everything. We learned in multiplication that it doesn't matter which order you multiply things in. I can just rearrange. We're just going and multiplying 2 times x times y squared times x to the 4th times y squared times 3 times x squared times y squared. I can rearrange this, and I will rearrange it, so that it's in a way that's easy to simplify. I can multiply 2 times 3, and then I can worry about the x terms. Let me do it in this color. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | We're just going and multiplying 2 times x times y squared times x to the 4th times y squared times 3 times x squared times y squared. I can rearrange this, and I will rearrange it, so that it's in a way that's easy to simplify. I can multiply 2 times 3, and then I can worry about the x terms. Let me do it in this color. Then I have times x times x to the 4th times x squared. Then I have to worry about the y terms. Times y squared times another y squared times another y squared. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Let me do it in this color. Then I have times x times x to the 4th times x squared. Then I have to worry about the y terms. Times y squared times another y squared times another y squared. What are these equal to? 2 times 3, you knew how to do that. That's equal to 6. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Times y squared times another y squared times another y squared. What are these equal to? 2 times 3, you knew how to do that. That's equal to 6. What is x times x to the 4th times x squared? One thing to remember is x is the same thing as x to the 1st power. Anything to the 1st power is just that number. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | That's equal to 6. What is x times x to the 4th times x squared? One thing to remember is x is the same thing as x to the 1st power. Anything to the 1st power is just that number. 2 to the 1st power is just 2. 3 to the 1st power is just 3. What is this going to be equal to? |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | Anything to the 1st power is just that number. 2 to the 1st power is just 2. 3 to the 1st power is just 3. What is this going to be equal to? This is going to be equal to, we have the same base, x, we can add the exponents. x to the 1 plus 4 plus 2 power. I will add it in the next step. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | What is this going to be equal to? This is going to be equal to, we have the same base, x, we can add the exponents. x to the 1 plus 4 plus 2 power. I will add it in the next step. Then on the y's, this is times y to the 2 plus 2 plus 2 power. What does that give us? That gives us 6x to the 7th power, y to the 6th power. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | I will add it in the next step. Then on the y's, this is times y to the 2 plus 2 plus 2 power. What does that give us? That gives us 6x to the 7th power, y to the 6th power. I will leave you with something you might already know. It's pretty interesting. That's the question of what happens when you take something to the 0th power. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | That gives us 6x to the 7th power, y to the 6th power. I will leave you with something you might already know. It's pretty interesting. That's the question of what happens when you take something to the 0th power. If I say 7 to the 0th power, what does that equal? I will tell you right now, this might seem very counterintuitive, this is equal to 1. 1 to the 0th power is also equal to 1. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | That's the question of what happens when you take something to the 0th power. If I say 7 to the 0th power, what does that equal? I will tell you right now, this might seem very counterintuitive, this is equal to 1. 1 to the 0th power is also equal to 1. Anything to the 0th power, any non-zero number to the 0th power is going to be equal to 1. Just to give you a little bit of intuition on why that is, think about it this way. 3 to the 1st power, let me write the powers. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | 1 to the 0th power is also equal to 1. Anything to the 0th power, any non-zero number to the 0th power is going to be equal to 1. Just to give you a little bit of intuition on why that is, think about it this way. 3 to the 1st power, let me write the powers. 3 to the 1st, 2nd, 3rd, we'll just do it with the number 3. 3 to the 1st power is 3, I think that makes sense. 3 to the 2nd power is 9. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | 3 to the 1st power, let me write the powers. 3 to the 1st, 2nd, 3rd, we'll just do it with the number 3. 3 to the 1st power is 3, I think that makes sense. 3 to the 2nd power is 9. 3 to the 3rd power is 27. We're trying to figure out what should 3 to the 0th power be. Think about it, every time you decrement the exponent, every time you take the exponent down by 1, you're dividing by 3. |
Exponent properties involving products Numbers and operations 8th grade Khan Academy.mp3 | 3 to the 2nd power is 9. 3 to the 3rd power is 27. We're trying to figure out what should 3 to the 0th power be. Think about it, every time you decrement the exponent, every time you take the exponent down by 1, you're dividing by 3. To go from 27 to 9, you divide by 3. To go from 9 to 3, you divide by 3. To go from this exponent to that exponent, maybe we should divide by 3 again. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | The king's advisor, Arbegla, is watching all of this discourse between you, the king, and the bird. And he's starting to feel a little bit jealous, because he's supposed to be the wise man in the kingdom, the king's closest advisor. So he steps in and says, OK, if you and this bird are so smart, how about you tackle the riddle of the fruit prices? And the king says, yes, that is something that we haven't been able to figure out, the fruit prices. Arbegla tell them the riddle of the fruit prices. And so Arbegla says, well, we want to keep track of how much our fruit costs, but we forgot to actually log how much it costs when we went to the market. But we know how much in total we spent. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | And the king says, yes, that is something that we haven't been able to figure out, the fruit prices. Arbegla tell them the riddle of the fruit prices. And so Arbegla says, well, we want to keep track of how much our fruit costs, but we forgot to actually log how much it costs when we went to the market. But we know how much in total we spent. We know how much we got. We know that one week ago, when we went to the fruit market, we bought two pounds of apples and one pound of bananas. And the total cost that time was $3. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | But we know how much in total we spent. We know how much we got. We know that one week ago, when we went to the fruit market, we bought two pounds of apples and one pound of bananas. And the total cost that time was $3. So there was $3 in total cost. And then when we went the time before that, we bought six pounds of apples and three pounds of bananas. And the total cost at that point was $15. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | And the total cost that time was $3. So there was $3 in total cost. And then when we went the time before that, we bought six pounds of apples and three pounds of bananas. And the total cost at that point was $15. So what is the cost of apples and bananas? So you look at the bird. The bird looks at you. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | And the total cost at that point was $15. So what is the cost of apples and bananas? So you look at the bird. The bird looks at you. The bird whispers into the king's ear. And the king says, well, the bird says, well, just start defining some variables here so we can express this thing algebraically. So you go about doing that. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | The bird looks at you. The bird whispers into the king's ear. And the king says, well, the bird says, well, just start defining some variables here so we can express this thing algebraically. So you go about doing that. What we want to figure out is the cost of apples and the cost of bananas per pound. So we set some variables. So let's let A equal the cost of apples per pound. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | So you go about doing that. What we want to figure out is the cost of apples and the cost of bananas per pound. So we set some variables. So let's let A equal the cost of apples per pound. And let's let B equal the cost of bananas per pound. So how could we interpret this first information right over here? 2 pounds of apples and a pound of bananas cost $3. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | So let's let A equal the cost of apples per pound. And let's let B equal the cost of bananas per pound. So how could we interpret this first information right over here? 2 pounds of apples and a pound of bananas cost $3. Well, how much are the apples going to cost? Well, it's going to cost 2 pounds times the cost per pound times A. That's going to be the total cost of the apples in this scenario. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | 2 pounds of apples and a pound of bananas cost $3. Well, how much are the apples going to cost? Well, it's going to cost 2 pounds times the cost per pound times A. That's going to be the total cost of the apples in this scenario. And what's the total cost of the bananas? Well, it's 1 pound times the cost per pound. So you're just going to have B. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | That's going to be the total cost of the apples in this scenario. And what's the total cost of the bananas? Well, it's 1 pound times the cost per pound. So you're just going to have B. That's going to be the total cost of the bananas, because we know we bought 1 pound. So the total cost of the apples and bananas are going to be 2A plus B. And we know what that total cost is. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | So you're just going to have B. That's going to be the total cost of the bananas, because we know we bought 1 pound. So the total cost of the apples and bananas are going to be 2A plus B. And we know what that total cost is. It is $3. Now let's do the same thing for the other time that we went to the market. 6 pounds of apples, the total cost is going to be 6 pounds times A dollars per pound. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | And we know what that total cost is. It is $3. Now let's do the same thing for the other time that we went to the market. 6 pounds of apples, the total cost is going to be 6 pounds times A dollars per pound. And the total cost of bananas is going to be, well, we bought 3 pounds of bananas. And the cost per pound is B. And so the total cost of the apples and bananas, this scenario, is going to be equal to $15. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | 6 pounds of apples, the total cost is going to be 6 pounds times A dollars per pound. And the total cost of bananas is going to be, well, we bought 3 pounds of bananas. And the cost per pound is B. And so the total cost of the apples and bananas, this scenario, is going to be equal to $15. So let's think about how we might want to solve it. We could use elimination. We could use substitution. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | And so the total cost of the apples and bananas, this scenario, is going to be equal to $15. So let's think about how we might want to solve it. We could use elimination. We could use substitution. Whatever we want. We could even do it graphically. Let's try it first with elimination. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | We could use substitution. Whatever we want. We could even do it graphically. Let's try it first with elimination. So the first thing I might want to do is maybe I want to eliminate the A variable right over here. So I have 2A over here. I have 6A over here. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | Let's try it first with elimination. So the first thing I might want to do is maybe I want to eliminate the A variable right over here. So I have 2A over here. I have 6A over here. So if I multiply this entire white equation by negative 3, then this 2A would become a negative 6A. And then it might be able to cancel out with that. So let me do that. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | I have 6A over here. So if I multiply this entire white equation by negative 3, then this 2A would become a negative 6A. And then it might be able to cancel out with that. So let me do that. Let me multiply this entire equation, the entire equation, times negative 3. So negative 3 times 2A is negative 6A. Negative 3 times B is negative 3B. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | So let me do that. Let me multiply this entire equation, the entire equation, times negative 3. So negative 3 times 2A is negative 6A. Negative 3 times B is negative 3B. And then negative 3 times 3 is negative 9. And now we can essentially add the two equations, or add the left side of this equation to the left side of that, and the right side of this equation to the right side of that. We're essentially adding the same thing to both sides of this green equation, because we know that this is equal to that. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | Negative 3 times B is negative 3B. And then negative 3 times 3 is negative 9. And now we can essentially add the two equations, or add the left side of this equation to the left side of that, and the right side of this equation to the right side of that. We're essentially adding the same thing to both sides of this green equation, because we know that this is equal to that. So let's do that. Let's do it. So on the left-hand side, 6A and 6A cancel out. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | We're essentially adding the same thing to both sides of this green equation, because we know that this is equal to that. So let's do that. Let's do it. So on the left-hand side, 6A and 6A cancel out. But something else interesting happens. The 3B and the 3B cancels out as well. So we're just left with 0 on the left-hand side. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | So on the left-hand side, 6A and 6A cancel out. But something else interesting happens. The 3B and the 3B cancels out as well. So we're just left with 0 on the left-hand side. And on the right-hand side, what do we have? 15 minus 9 is equal to 6. So we get this bizarre statement. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | So we're just left with 0 on the left-hand side. And on the right-hand side, what do we have? 15 minus 9 is equal to 6. So we get this bizarre statement. All of our variables have gone away, and we're left with this bizarre, nonsensical statement that 0 is equal to 6, which we know is definitely not the case. So what's going on over here? What's going on? |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | So we get this bizarre statement. All of our variables have gone away, and we're left with this bizarre, nonsensical statement that 0 is equal to 6, which we know is definitely not the case. So what's going on over here? What's going on? And then you say, what's going on? And you look at the bird, because the bird seems to be the most knowledgeable person in the room, or at least the most knowledgeable vertebrate in the room. And so the bird whispers into the king's ear. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | What's going on? And then you say, what's going on? And you look at the bird, because the bird seems to be the most knowledgeable person in the room, or at least the most knowledgeable vertebrate in the room. And so the bird whispers into the king's ear. And the king says, well, he says that there's no solution, and you should at least try to graph it to see why. And so you say, well, the bird seems to know what he's talking about. So let me attempt to graph these two equations and see what's going on. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | And so the bird whispers into the king's ear. And the king says, well, he says that there's no solution, and you should at least try to graph it to see why. And so you say, well, the bird seems to know what he's talking about. So let me attempt to graph these two equations and see what's going on. And so what you do is you take each of the equation. And when you graph it, you like to put it in kind of the y-intercept form or slope-intercept form. And so you do that. |
Inconsistent systems of equations Algebra II Khan Academy.mp3 | So let me attempt to graph these two equations and see what's going on. And so what you do is you take each of the equation. And when you graph it, you like to put it in kind of the y-intercept form or slope-intercept form. And so you do that. So you say, well, let me solve both of these for b. So if you want to solve this first equation for b, you just subtract 2a from both sides. If you subtract 2a from both sides of this first equation, you get b is equal to negative 2a plus 3. |
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