Hey guys, welcome to Algebra 1. Today we solve problems involving the power of a quotient. Your knowledge of exponents and products and quotients will take you far in this lesson. Ready to get going? Let's start.

(Describer) With a stylus, she presses button on a screen.

All right, let's get the pen here. Okay, knowing what you do about exponents, let's expand this: 7 halves to the 3rd power.

(Describer) ...on the screen.

I'm raising 7 halves, that quotient, to the 3rd power. That means I'm multiplying it by itself 3 times. I could expand this to 7 halves times 7 halves

(Describer) She writes.

times 7 halves, right? Just expand it out-- what exactly that means. Now to make it simpler, I'm going to just put all of those 7s together for my numerator-- 7 times 7 times 7. And I'm going to do the same thing with my denominator: 2 times 2 times 2, right? Didn't change the meaning at all. Just representing it differently. Now if I wanted to simplify this and write it and represent it a different way, I know that 7 times 7 times 7 is the same as 7 to the 3rd, and that 2 times 2 times 2 is the same as 2 to the 3rd. That means that 7 halves to the 3rd power is the same as 7 to the 3rd divided by 2 to the 3rd. Okay, keep going. We establish a pattern that's going to help us understand this rule. So 4/9 to the 5th-- I'm going to use that same process to understand this one. So 4/9 to the 5th power, that's the same as 4/9 times 4/9 times 4/9 times 4/9 times 4/9. Right? the product of 5 "4/9ths." Now I'm going to group those numerators together. Let me get one big fraction bar. I write that as 4 times 4 times 4 times 4 times 4, right? I got the product of those 5 4s together with my numerator, and same for the denominator: 9 times 9 times 9 times 9 times 9, right? Now if I wanted to simplify that and bring my exponents back, I have the product of 5 4s in my numerator. That's 4 to the 5th up top. And I have the product of 5 9s in the denominator. That's 9 to the 5th on the bottom. So 4/9 to the 5th, the problem we originally started with, could be written as 4 to the 5th over 9 to the 5th. You starting to notice the pattern yet? Let's try another one. All right, c over d to the 4th power. Applying that same process, I know I could expand this and write it as: c over d times c over d times c over d times c over d. Right, okay. Then I'm going to group all those cs together. So I'd have c times c times c times c over d times d times d times d, right? Then I'm going to represent that using exponents to write it simpler. The product of 4 cs in my numerator, that's c to the 4th. And the product of those 4 ds in the denominator, that's d to the 4th, right? That c over d to the 4th power could be written as c to the 4th over d to the 4th. Do you see what's going on? It's really like you're distributing your exponent throughout the terms of what's inside your parentheses. In this problem, c over d raised to the 4th power means the same as c to the 4th power over d to the 4th power. That's the rule. Let's look at it written algebraically. Get that pointer tool back, okay. So for the power of a quotient, anytime we're dividing a quotient and we're raising it to the power, you can split it. Raise your numerator to whatever your power is and raise your denominator to whatever your power is. It's like you're distributing the exponent throughout your quotient.

(female describer) Title: Power of a quotient--

(Describer) Title: Power of a Quotient. A over m to the r power equals a to the r power over m to the r power.

a over m to the r power equals a to the r power over m to the r power. Let's apply this; let's try a couple of problems. Our first example, we have x over 8 raised to the 2nd power. I'm going to apply that rule-- get the pen back-- for the power of a quotient. I raise my numerator to the 2nd power. That would become just x to the 2nd power. I'm going to raise my denominator to the 2nd power-- just 8 to the 2nd power. Because I don't know what x is, I can't make that value any more simple, but I can represent 8 squared as 64, okay? That means that your final answer for this problem would be x squared over 64. We need to raise our numerator and then raise our denominator to whatever that power is, and you've applied the rule. Okay, let's try another one. Let's click off of that. It's going to start moving it around if I don't. So p over 4 to the 3rd power-- so apply that rule. Raise your numerator to the 3rd power. So p to the 3rd power. And I'm going to raise my denominator to the 3rd power, so 4 to the 3rd power, okay? I don't know what p is. I can't represent that another way. It's just p to the 3rd. I can't represent it simpler. Now 4 to the 3rd-- Let's come off to the side. I'm doing mental math, but you can use a calculator. You can get it that way. 4 to the 3rd; that's 4 times 4 times 4. So 4 times 4, that's 16. And then 16 times 4, that's 64. I'm going to put 64 in my denominator, and I'm all done. The answer to that problem is p to the 3rd over 64. All right, now it's your turn. Click off for a second; there we go. Take a look at these. Pause the tape and try them. See how you do. When you're ready to compare answers, press play.

(female describer) Title: Simplify.

(Describer) Title: Simplify. Number One: y over 9, to the second power. Number Two: b over 2, to the fourth power.

1. y over 9 to the 2nd power. 2. b over 2 to the 4th power.

(teacher) All right, ready to check? Here we go. For the first one, y over 9 to the 2nd power. You should have y squared over 81. And b over 2 to the 4th power, that simplifies to: b to the 4th over 16. If you want to see how I did those, stay with me. Let me get my pen back, okay. When I did the first one, I just applied that rule. I raised y to the 2nd power and I raised 9 to the 2nd power, all right? So, y squared-- couldn't do any more with that. And then 9 squared-- I know that's 9 times 9, so that's 81. That's how I got that first one, okay? Now let's check that second one. Here I had b over 2 to the 4th power. I raised b to the 4th and I raised 2 to the 4th. I couldn't do anything with that b to the 4th. That stayed as it was. Now 2 to the 4th-- and come off to the side. That's the same as 2 times 2 times 2 times 2. I know 2 times 2 is 4, 4 times 2 is 8, and 8 times 2 is 16. That's how I got that denominator, and then I was all done. All righty. Now those were some basic ones. We're going to step it up, but we're going to apply the same rule, okay? Take a look at this. It looks overwhelming, but we'll break it up. We apply the same rule.

(female describer) 3x to the 7th power, y squared

(Describer) Three x to the seventh power y-squared over 5z to the fourth power, all cubed.

over 5z to the 4th power, all cubed.

(teacher) I see I have my numerator, that 3x to the 7th, y squared over my denominator, that 5z to the 4th. I'm raising that quotient to the 3rd power. I'm going to take extra steps to break it down further to make sure you're completely clear with everything. Okay? Let's break this up. I'm going to raise my numerator to the 3rd power, right? We know from what we were just doing

(Describer) She writes.

when we apply that power of a quotient rule, that's-- I could expand it this way.

(Describer) 3x to the seventh, y-squared in the numerator.

I raise that to the 3rd. And the same to my denominator. I'm going to raise my denominator to that same power.

(Describer) 5z to the fourth in the denominator.

With me so far? Broke it up. We raise our numerator to the 3rd power, and we need to raise our denominator to the 3rd power. Now-- Now you apply a different rule from your rules of exponents. Now I have a product that I need to raise to a power. Remember, when you do that, you raise each individual piece to that 3rd power. It's all we need to do. I'm going to raise this 3 to the 3rd power. I'm going to raise that x to the 7th to the 3rd power... and I'm going to raise that y squared to the 3rd power. Okay? All I did there was take each individual piece in my numerator and raise it to the 3rd power. I do the same thing with the denominator. I need to raise 5 to the 3rd power, and then I need to raise z to the 4th to the 3rd power. Okay? Now that I'm looking at it like this, I need to keep simplifying and see what this actually ends up coming out to. So, 3 to the 3rd-- I know that's 3 times 3 times 3. 3 times 3, that's 9. And 9 times 3 is 27. I know that 3 to the 3rd is 27. Look at this x to the 7th. Remember your rule about raising a power to a power? All you do is multiply those exponents together. That's the shortcut. If I raise x to the 7th to the 3rd power, I just need to multiply those exponents together. 7 times 3, that's 21. That will become x to the 21st. The same thing here; I'm raising a power to another power. I need to multiply these exponents together-- 2 times 3... That's 6. So y to the 6th, okay? Follow that same process with your denominator. You have 5 to the 3rd, right? So 5 times 5, that's 25. 25 times 5, that's 125. I have 125. I have z to the 4th. I'm raising that to the 3rd power, so my rule, raising a power to a power-- I multiply exponents together. 4 times 3, that's 12. So, z to the 12th power. You can take it no further. You are done, okay? That's how you do that. It's more complicated, but you'll get it. We'll do more. Just break it up. And you may not want that middle step because you'll get comfortable, but I want to do it to make sure you get comfortable. Break up your quotient, raise it to whatever power you're being asked to, and apply the rule of exponent you need to simplify it. Let's try another one. Let's keep going. Let's make sure I have enough room here. Let's scoot that over a bit, okay? We have 9s squared, t to the 5th over 4p cubed q to the 8th, all raised to the 2nd power. All right. I break it up first. I split up my numerator and my denominator. I know that I need to raise 9s squared, t to the 5th to the 2nd power, and I need raise 4p to the 3rd, q to the 8th to the 2nd power. Right? Okay. I have to apply my rule about raising a product to a power. I raise each individual piece of my product to that power. That means I'd have 9 squared times s squared to the 2nd power, and then I need to raise t to the 5th to the 2nd power. Okay? I've handled the numerator. Let's go to the denominator. I need to raise each individual piece to the 2nd power. So, 4 squared-- handled that. I have p to the 3rd squared. Then I have q to the 8th squared. I'll write q the same way so you don't think that's a 9. Let's just clean this up. I know that 9 squared is 81, so I've handled that. I apply my rule about raising a power to a power. Multiply those exponents together-- 2 times 2, that's 4. So, s to the 4th. Apply that same rule here, right? So, t to the 5th raised to the 2nd power. 5 times 2, that's 10. So t to the 10th. I've handled the numerator. Now on to the denominator. I know 4 squared, that's 16, right? Now I have p cubed raised to the 2nd power. So multiply those exponents together. 3 times 2, that's 6. So p to the 6th. Then I have q to the 8th raised to the 2nd power. Apply the same rule-- 8 times 2, that's 16. So q to the 16th. All right? You're all done. Got it all simplified, okay? Getting a little more comfortable? Let's try another one. OK, 2c to the 5th, d to the 6th, f to the 4th over 3g to the 7th, t squared-- We raise all that to the 3rd power. It is a mouthful. Let's get going, though. I'm going to split it up. I'm going to raise my numerator to the 3rd power. So, 2c to the 5th, d to the 6th, f to the 4th. I'm going to cube that. Then same for my denominator. 3g to the 7th, t squared, and I'm going to cube that, right? I'm going to come down to get more room. I'm going to raise each individual piece of my numerator to the 3rd power. So I have 2 to the 3rd times c to the 5th to the 3rd. Then I've got to handle that d to the 6th, so times d to the 6th to the 3rd, times f to the 4th to the 3rd. Okay? Then for my denominator, I have 3 cubed, All right? And I'm going to multiply that times g to the 7th to the 3rd power. Okay, then I'm going to multiply that times t squared to the 3rd power. Okay? All right. Now that I have it all broken up, I need to handle simplifying this. Let's find some little more room for us to work with. Get rid of this because you've handled this part. I'm going to get it out of our way

(Describer) She erases her first step in breaking things down.

and I'm going to bring up what we just did.

(Describer) She drags it up beside the original quotient.

All right, now we got a little more room to work here. Let's clean up that eraser. Let's bring up to our 2; it disappeared. All right, here we go. So now, 2 cubed. I know that's 8-- 2 times 2 times 2. Then I have c to the 5th raised to the 3rd power. I know I need to multiply my exponents together. So 5 times 3, that's 15. I have d to the 6th raised to the 3rd power. Multiply your exponents there. So, 6 times 3, that's 18. I have f to the 4th to the 3rd power. I multiply my exponents there as well. So 4 times 3, that's 12. So, f to the 12th, all righty? All over-- let's handle this denominator. 3 cubed, we know that's 27-- 3 times 3 times 3. g to the 7th raised to the 3rd power, multiply your exponents together there. Seven times 3, that's 21. So, g to the 21st. Then we have t squared raised to the 3rd power. Multiply your exponents here-- 2 times 3, that's 6. So, t to the 6th. And you're all done. All right, I believe it's your turn. Go ahead and pause me. Take a few minutes; work through these problems. When you're ready to compare answers, press play.

(female describer) Title: Simplify.

(Describer) Title: Simplify. Number One: 7h to the fourth, k to the fifth, over 8j to the ninth, m to the sixth, all squared. Number Two: 4a to the tenth, b-squared, c to the seventh, over 3d to the fifth, e-squared, all cubed.

1. 7h to the 4th, k to the 5th over 8j to the 9th, m to the 6th, all squared. 2. 4a to the 10th, b squared, c to the 7th over 3d to the 5th, e squared, all cubed.

(teacher) Ready to check? Let's check. For the first one, we had 7h to the 4th, k to the 5th over 8j to the 9th, m to the 6th, and we were squaring that quotient. You should've got 49h to the 8th, k to the 10th over 64j to the 18th, m to the 12th, okay? For the second one, 4a to the 10th, b squared, c to the 7th over 3d to the 5th, e squared. We were raising that quotient to the 3rd power. Your answer should've been-- there we go-- 64a to the 30th, b to the 6th, c to the 21st over 27d to the 15th, e to the 6th. If you want to see how I did either of these, here we go. Let's get the pen back. All righty. For this one, we applied our same rule we've been applying.

(Describer) ...for the first one.

I do a little shifting here because I'm running out of space. There we go. I broke our quotient up. I raised the numerator to the 2nd power. So, 7h to the 4th, k to the 5th squared. And I raised the denominator to the 2nd power-- 8j to the 9th, m to the 6th squared. Then I broke up those products even further. For the numerator, I raised 7 to the 2nd power. Then I needed to raise h to the 4th to the 2nd power. Then I needed to raise k to the 5th to the 2nd power. Then I'd handle my numerator. Same process for the denominator. I need to raise 8 to the 2nd power. Then I need to raise j to the 9th to the 2nd power. Then I needed to raise m to the 6th to the 2nd power. All righty. Then we simplify this. So, 7 squared, that's 49. We're raising a power to a power, so multiply those exponents-- 4 times 2 is 8. So, h to the 8th. Same thing here-- multiply our exponents together right there. So 5 times 2, that's 10. So, k to the 10th. For the denominator, 8 squared-- I know that's 64. We have j to the 9th raised to the 2nd power, so multiply those exponents together. We've got j to the 18th. We have m to the 6th raised to the 2nd power-- 6 times 2 is 12, so m to the 12th. All righty? And that's how we got that first answer. Now if you need to see the second one, here we go. I followed my same process. I broke it up to start. For the numerator, I had 4a to the 10th, b squared, c to the 7th, and I raised that to the 3rd, that whole product. Then for the denominator, I had 3d to the 5th, e squared, and I raised that whole product to the 3rd power, right? Then we just broke up those individual products. And I will come down here. So I have 4 to raise to the 3rd power in the numerator, times a to the 10th raised to the 3rd power... times b squared raised to the 3rd power... times c to the 7th raised to the 3rd power. All righty. We handled the numerator. Now for the denominator, we had 3 to the 3rd. We have d to the 5th that we raise to the 3rd power... And we have e squared that we need to raise to the 3rd power. This was one of those long ones. Let's make more room here. Let's get rid of that. We've taken care of that. Let's bring our current work up. We made a little more room to work here. All righty. Now 4 cubed, I think we did that earlier-- 4 times 4 times 4, that's 64. We've got that first bit, and then your power-to-a-power rule. Multiply your exponents there-- 10 times 3, that's 30. So a to the 30th. Keep applying that rule-- 2 times 3 for those exponents, that's 6. So b to the 6th. And for c-- 7 times 3, that's 21. So c to the 21st. Okay? And the same process for our denominator. 3 to the 3rd, that's 27. Power-to-a-power rule right here-- 5 times 3, that's 15. So, d to the 15th. Then here we have e squared to the 3rd power. Multiply those exponents there-- 2 times 3 is 6. So, e to the 6th. That is how I got the second one. I hope that you're feeling more comfortable with these problems and that your knowledge of exponents took you through this. See you soon. Bye, guys.

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