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Welcome to Algebra I: Power of a Product

22 minutes

Hey, guys. Welcome to Algebra 1. Today we're going to get started solving problems about a power of a product. Use all you know about exponents and patterns because they're going to help you get through this lesson. Ready to get started?

(Describer) She uses a stylus on a screen.

Let's go. All right, to help you with this I need to set up a pattern for you. Bear with me, but it's going to help you understand how you solve these kind of problems. 4Y raised to the 3rd power, that product. What does that actually mean if we expand it out? I'm multiplying 4Y by itself three times. So 4Y times 4Y... times 4Y. Right? The commutative property of multiplication tells me I can multiply things in any order that I feel like, right? I'm going to rearrange this multiplication. I'll bring all those coefficients out front-- all those 4's--and I'll group all those Y's together. Okay? I'm going to represent this as 4 times 4 times 4 and Y times Y times Y. Right? Okay, now I'm going to simplify this. I see this part, right here. I've got those three 4's multiplied together. Then here I've got those three Y's multiplied together. I could represent that first chunk, that 4 times 4 times 4, as 4 to the 3rd and I could represent that second chunk, Y times Y times Y, as Y to the 3rd. That means that 4Y raised to the 3rd power, it means the same thing as 4 to the 3rd times Y to the 3rd, right? Okay, keep that in mind and keep going with me. Do this a few more times. So this one's a little different, but we'll apply our same rule, that same process that we just did. So 10B cubed raised to the 2nd power. That means I'm multiplying 10B cubed by itself just one time. I would have 10B cubed... times 10B cubed. Right? If I expanded that out, okay? I'm going to rearrange things like we did before. I'll bring my 10s out front and put my B cubes to the back. I'm going to rearrange this as 10 times 10 times B to the 3rd times B to the 3rd, right? Just rearranged the order of the multiplication. Didn't change the meaning of the problem, okay? Now I'm going to look at these in groups. I'll throw those parentheses back in there to simplify this. If I'm multiplying 10 by itself one time, that's the same thing as 10 squared because you just raise 10 to the 2nd power. Multiply it by itself once. Then over here I have B to the 3rd times B to the 3rd, right? Isn't that the same thing as squaring B to the 3rd, just multiplying it by itself one time? I could represent it like this: I've got 10 times 10, which I'm representing as 10 squared, and I've got B cubed times B cubed, which I'm representing as B cubed squared. That means that 10B to the 3rd raised to the 2nd power, that problem we started with, could be written just as this. 10 squared times B cubed to the 3rd, okay?

(Describer) B-cubed squared.

Stay with me, keep going. 3X squared, that whole quantity, raised to the 4th power. All right, so let's think about what this means. You're multiplying 3X squared by itself 4 times. So you would have 3X squared... times 3X squared... times 3X squared... times 3X squared. Okay? Now I'll do that same rearranging. I'll bring all those 3's out front and put all those variables at the back. All right, so I've got four 3's I'm multiplying together, so one, two, three, four. Now, I pulled each of those 3's out front. And then I have X squared that I'm multiplying by itself four times. So I'll bring all those together at the back. Okay, now I'm going to wrap parentheses around those different groups just to help me zone in on them. All right, so I have 3 times 3 times 3 times 3. I could represent that as 3 to the 4th, right? Because you're multiplying 3 by itself 4 times. Now, here you have X squared, times X squared, times X squared, times X squared. You're multiplying X squared by itself four 4 times. So you could represent that part as X squared, that quantity, raised to the 4th power. That means that 3X squared, that whole bit, raised to the 4th power is the same as 3 to the 4th times X squared to the 4th. Are you noticing anything yet? Let's keep going. All right, let's look at those three answers together, and see if you notice the pattern. See if you notice what's going on. Okay, when we did 4Y, that quantity cubed, we ended up simplifying it as 4 to the 3rd times Y to the 3rd. Right? And when we had 10B cubed raised to the 2nd power, we represented that as 10 squared times... Let me write it just like we wrote it. I want you to really look at them all in the same way. Okay. I'm going to write 10 squared times-- I've got to do B cubed squared. I was trying it with the parentheses, but I can't. And then we had 3X squared raised to the 4th power, right? We had 3 to the 4th and X squared to the 4th. Now look at those all together, even pause me for a bit. See if you notice the pattern. There's something going on with the exponents that we start with in our original problem and the exponents that we end up with in the solution. Pause me for a bit. See if you can figure out that pattern. All right, did you notice? Let's look at it together. In that first one, we had 4Y raised to the 3rd power, and then for our answer we had 4 to the 3rd times Y to the 3rd. We distributed that exponent throughout that product inside. We broke it up, raised 4 to the 3rd power and then raised that Y to the 3rd power. If you look at that second example, that 10B to the 3rd squared, it's like we took that 10 and raised it to the 2nd power and then we took that B cubed and raised it to the 2nd power. It's like we took both parts and raised them to the power of our exponent on the outside. Look at that last one. 3X squared raised to the 4th power. We took that 3 and raised it to the 4th power for this part, and we took that X squared, and raised it to the 4th power for this part. It's like you're distributing your exponent throughout the terms inside your parentheses. That rule is known as... the power of a product. If you have a product that you have to raise to some power, just split the factors of your product up and raise them to that power initially. That's a shortcut to solving these problems that involve a power of a product. Let's apply this rule and solve a few more.

(Describer) Titles: Power of a Product. Open parenthesis a-m close parenthesis, to the r power, equals a to the r power times m to the r power.

(female narrator) Power of a Product:

(AM) to the R power equals A to the R power times M to the R power. All right, let's get that pen back. Okay. I have 5B cubed C, that quantity, and I'm raising it to the 2nd power. Instead of expanding it out and finding the pattern, I'll use the rule we just figured out. I'll take each of these terms, each of those bits, separately, and I'll raise them to the 2nd power. All right? Watch me work. I'm going to take that 5 and raise it to the 2nd power, so 5 squared, then I'm going to take this B cubed and raise it to the 2nd power. So B cubed, squared. Then I'm going to take that C and raise it to the 2nd power, okay? All right, so now let's simplify this. Now that you know the rule, we're going to go ahead and simplify 5 squared, okay? We know that 5 times 5 is what this represents, right? So 5 squared, that's 5 times 5. So that's 25, right? I know that that first part just simplifies to 25. Now for this one, that B cubed raised to the 2nd power, let's come off to the side. B cubed raised to the 2nd power. We'll use our rule about raising a power to a power. Remember when you raise a power to a power, you just multiply your exponents together. I know 3 times 2, that's 6. That means that that simplifies as B to the 6th, right? I know the second part, right here, I can rewrite as B to the 6th. You see I'm dropping my dots? I'm getting ready to write the answer. You could keep writing those multiplication dots, though. Then I see C raised to the 2nd power. I'll come off to the side. So C raised to the 2nd power. Did not write that like it was. Let's write it just like we started. Okay, so C raised to the 2nd power. Okay, whenever you don't see an exponent, remember there's a 1 there, it's an invisible 1. Same thing, power to a power, it's like you're multiplying your exponents together, that is what you're doing. So 1 times 2 is 2. So C raised to the 2nd power is C squared. You probably already knew that anyway, okay? Then you're all done. That would break right on down to 25B to the 6th, C to the 2nd power, okay? Let's try another one just to make sure you got the hang of it. So for this one, we have 4X to the 6th, Y squared, raised to the 3rd power, right? Just use our rule. We're going to raise each of these parts to the 3rd power, like we're distributing the exponent, almost. All right, so I'd have 4 to the 3rd power. I'm going to raise everything to that exponent on the outside. That's times X to the 6th, raised to the 3rd power, right? Then I'd have Y squared raised to the 3rd power. Now I'm just going to come off to the side and do a little work to simplify all this. All right. So 4 to the 3rd power. I know that that's 4 times 4, times 4. So 4 times 4, that's 16. And 16 times 4 is 64. If you don't want to do mental math like I'm doing right now, you could pull out a calculator and do 4 to the 3rd and get 64. We've got that first part. Now we have X to the 6th raised to the 3rd power. We've got to handle this. Remember, we use our rule about applying a power to a power. Just multiply your exponents together. So 6 times 3, that's 18. That would be X to the 18th. And then I have-- I'll write over here, I'm running out of room-- Y squared raised to the 3rd power. So our rule about raising a power to a power, just multiply your exponents together. So 2 times 3, that's 6. This would be Y to the 6th, for that last bit. And you're all done.

(Describer) 64x to the 18th, y to the 6th.

(female narrator) 64X to the 18th, Y to the 6th. Just distribute that exponent throughout and you're all done. All right. Okay, one more together. So I have 2MN to the 5th raised to the 4th power. So I'm going to just distribute that exponent-- I'm thinking about it like that--throughout. So I have 2 raised to the 4th power. That's 2 to the 4th. Then I have M raised to the 4th power. So times M to the 4th. And I have N raised to the 5th power-- I'm sorry, N to the 5th raised to the 4th power. N to the 5th, raised to the 4th. All right, now let's simplify this. So I'm going to come off to the side. 2 to the 4th, that's 2 times 2, times 2, times 2, right? So 2 times 2 is 4. 4 times 2 is 8. 8 times 2 is 16. Okay? So I've simplified that part to 16. M to the 4th. I did a shortcut there. I did it differently than that one that had a 1 inside for the exponent. It'll save me work. It's just M to the 4th. There's a 1 there I don't see, so you're not changing anything when you raise that to the 4th power. Let me just show you, just in case you need it. There's a 1 there that I don't see. So when I multiply 1 times 4, it's just 4. You're raising it to that exponent on the outside, not multiplying it by a number that will change that exponent's value. Give yourself a shortcut when you run into that stuff. For this one, I have N to the 5th raised to the 4th power. Come off to the side. So N to the 5th to the 4th, right? We're raising a power to a power. Just multiply your exponents together. So 5 times 4, that's 20. So N to the 20th, for the last bit. And you're all done. You handled all three pieces of that. Okay? All right, let's keep going. It is now your turn, so pause, take a few minutes, and work your way through these problems. To compare answers with me, press play.

(Describer) Titles: Simplify the following expressions. Number One: 6g to the fourth, h to the third, all squared. Number Two: 3d to the 11th, p to the 6th, all to the third.

(female narrator) Simplify the following expressions. Number 1: 6G to the 4th, H to the 3rd, all squared. Number 2: 3D to the 11th, P to the 6th, all to the 3rd.

(instructor) All right, let's see how you did. 6G to the 4th, H to the 3rd, that quantity, raised to the 2nd power. You should have got 36G to the 8th, H to the 6th. 3D to the 11th, P to the 6th, that quantity cubed. You should have got 27D to the 33rd, P to the 18th, okay? If you need to see how I did any of those, let me show you. All right, so that first one. 6G to the 4th H cubed, that quantity raised to the 2nd power. Make sure I got my pen. I'm going to distribute that exponent throughout that product in my parentheses. I'm going to raise 6 to the 2nd power, I'm going to raise G to the 4th, to the 2nd power. Let's get that multiplication in there to break things up. And I'm going to raise H cubed to the 2nd power. Right? So then I'm going to come off to the side. I need to get 6 squared. Okay, 6 squared, that's 6 times 6. So that's 36. Okay? I'm going to replace 6 squared with 36. Then I have this piece, that G to the 4th, raised to the 2nd power. So G to the 4th, raised to the 2nd power. You're going to multiply your exponents together because you're applying your rule about raising a power to a power. Okay, so 4 times 2, that's 8. So you have G to the 8th, and then our last bit, that H cubed raised to the 2nd power. Same thing; apply your rule about raising a power to a power. Multiply your exponents together. 3 times 2, that's 6. So H to the 6th for the last bit, okay? That is how I got that first one. All right, if you need to see the second one, stay with me. Here we go. I'm going to distribute that exponent throughout what's inside my parentheses, basically what I'm doing. All right, so I have 3 raised to the 3rd power times D to the 11th, raised to the 3rd power times P to the 6th, raised to the 3rd power. Okay? Then just simplify this. Just keep breaking this down. So 3 to the 3rd-- come off to the side. All right, so that's 3 times 3 times 3. So 3 times 3, that's 9. 9 times 3 is 27. So I know 3 cubed would simplify to 27. Then I have D to the 11th, raised to the 3rd power. Come off to the side. D to the 11th, raised to the 3rd power. Follow your rule about raising a power to a power. Multiply your exponents together. So 11 times 3, that's 33. So that would be D to the 33rd power. Then we have our last piece, that P to the 6th raised to the 3rd power. Multiply your exponents together. So 6 times 3, that's 18. So This would be P to the 18th. And that's the last piece. All right, that is how I got that answer. All right. I hope you got a handle on the problems we did today, and that you saw how using your knowledge of exponents and patterns can help with the laws of exponents. All right, see you next time.

(Describer) Accessibility provided by the US Department of Education.

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To find a power of a product, students must find the power of each factor and then multiply. In other words, students can keep the exponent the same and multiply the bases. Part of the "Welcome to Algebra I" series.

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Runtime: 22 minutes

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