# 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.

(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:

(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.

(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.

## Media Details

Runtime: 22 minutes

Welcome to Algebra I
Episode 1
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Episode 2
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Episode 3
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Episode 5
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Episode 6
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Episode 7
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