# Welcome to Algebra I: Power of a Quotient

29 minutes

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

(female describer) Title: Simplify. 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.