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Welcome to Algebra I: Quotient of Powers

28 minutes

Hi, guys. Welcome to Algebra 1. Today we'll focus on the quotient of powers. You'll use what you know about exponents, division, patterns to help you work through these problems. Ready to get going? Let's go. Okay, to help you understand these, I'll take a little time going back to reducing fractions. Now, probably sometime in elementary school, you learned to reduce a fraction by finding the greatest common factor between the numerator and the denominator, then starting to cancel things out. So, you have a 15 in the numerator, a 20 in the denominator. Think about the factors of 20 and 15. Factors are the numbers that you multiply together to get your product. You may have different factors, like multiplication. For example, 20. I could get 20 by multiplying 1 times 20, 10 times 2, 5 times 4. But I want to consider the factors-- the one it has in common with 15. For example, I know I can get 15 by multiplying 5 times 3. I know 5 times 3 is 15, and I know that I can get 20 by multiplying 5 times 4. I picked those factors because they had the 5 in common-- the factors of 20 and of 15. You have a 5 in the numerator and a 5 in the denominator; you can cancel that out. Say, "Okay, then 15/20ths reduces to 3/4ths," right? You could just reduce that fraction without using a calculator. Keep that in mind as we work our way through these problems. Stay with me. You might be feeling a little "ugh" about these fractions, but we're going to work our way through it. So I see I have 32 over 28. Think about the factors 32 and 28 have in common. I want the greatest common factor. Knowing what you know about factors, I know that I could multiply 8 times 4, and that would give me 32. And I know that I could multiply 7 times 4, and get 28. So the greatest common factor between those two numbers is 4. Cancel out the 4s, and you could say, "Okay, then that fraction simplifies to 8/7ths." Right? Just reduce that fraction. Keep that in mind. Let's keep going. So there's some exponents in here. You have 2 to the 5th divided by 2 squared. Let's just expand those. Expand that numerator, expand that denominator. Look at it as a product of its factors, of a bunch of 2s. The numerator is 2 to the 5th, so that's the product of five 2s: 2 times 2 times 2 times 2 times 2, all right? In my denominator, I have 2 squared, so that's the product of two 2s, all right? Okay, so 2 times 2. Keep in mind what we just did with those fractions. Cancel out factors that they have in common. Cancel out a 2 in the numerator with a 2 in the denominator-- that would cancel out. Cancel out another 2 in the numerator with this 2 in the denominator, and that would just leave me with those three 2s up top. So, 2 times 2 times 2. If I wanted to simplify that, I could represent it as 2 to the 3rd power. So 2 to the 5th divided by 2 squared simplifies to 2 to the 3rd, okay? Keep doing a few more of these with me. You're going to notice the pattern. I've got 8 to the 6th divided by 8 to the 4th. That means in my numerator I have 8 multiplied by itself six times, okay? So 8 times eight times 8 times 8 times 8 times 8. Let's double check that. One, two, three, four, five--yep, six 8s. Then in my denominator, I have 8 to the 4th, so that's the product of four 8s. So 8 times 8 times 8 times 8, okay? I've expanded the numerator and denominator to look at it as a product of 8s. What can we cancel? You remember one 8 in the numerator will cancel out one 8 in the denominator. So I can cancel out that pair. That's another pair. That's another pair. That's another pair, all right? I ran out of 8s in the denominator, with two left in the numerator. 8 times 8, that's 8 squared. If you look at where we started, 8 to the 6th divided by 8 to the 4th is 8 squared, okay? Let's keep going. Are you picking up on the pattern here? I've got X to the 7th divided by X to the 3rd. What do we have on top? The product of seven Xs. So X times X times X times X times X times X times X. Double check that-- make sure we got seven. One, two, three, four, five, six, seven. So, we have X to the 7th on top. Denominator, we have X cubed. So that's X times X times X, okay? What do we do now? See what you can cancel out. An X in the numerator will cancel out with one in the denominator. Another pair cancels out, and another pair can cancel out. When I do that, I'm left with four Xs in my numerator, the product of four Xs. So X times X times X times X. That's X to the 4th power. So X to the 7th divided by X cubed equals X to the 4th. Okay? Let's keep going. Let's look at these three all together. When put together-- you may notice the pattern happening here. When we had 2 to the 5th divided by 2 squared, that answer was 2 to the 3rd. 8 to the 6th divided by 8 to the 4th, that was 8 squared. X to the 7th divided by X cubed, that was X to the 4th, okay? Take a minute. Pause me for a second. Look at these three problems, what we started with and what we ended with. Pay close attention to the exponents. Do you notice a relationship? Something's going on with the exponents that we start with and then the one that we end with. Did you notice it? Okay, I've got my pen. 5 minus 2, that's 3. 6 minus 4, that's 2. And 7 minus 3, that's 4. Taking the quotient of powers, the shortcut, because we've noticed this pattern, is just subtract the exponents that you start with. That's how we come upon this rule for the quotient of powers. For two terms with the same base-- here, it's a-- and you're finding the quotient, just subtract their exponents to get the answer. You won't have to write all those factors out, and cancel out. Just use this rule we discovered from using those problems to find a pattern. Let's start practicing with this rule. Let's get the pen back. X to the 18th divided by X to the 10th. I have a quotient of powers. All I need to do here-- my bases are both Xs-- is subtract the exponents. Here, 18 and 10 are my exponents, so 18 minus 10, that's 8. Notice it's the top exponent, subtract the bottom one, okay? So X to the 18th divided by X to the 10th, that's X to the 8th, and that's your answer. To apply that rule, subtract those exponents. Let's keep going. Y to the 9th divided by Y to the 4th, so same thing. Apply the rule, just subtract your exponents. Come off to the side. 9 minus 4, that's 5. This would simplify to Y to the 5th. If you show the product of those nine Ys and of those four Ys, and canceled out, you would end up with the product of five Ys. The answer is Y to the 5th. Okay? Let's keep moving. This one is a little different, but not too different. C to the 11th times D to the 15th divided by C to the 10th times D to the 8th. You have two different bases-- C and D. Just take them separately, like grouping your like terms together. For example, I would first handle that C to the 11th over C to the 10th. Apply my rule, and just subtract my exponents. So 11 minus 10, that's 1. That would simplify to C to the 1st. With an exponent that's 1, you don't have to write it. But I will just to reinforce that we just subtracted our exponents. So, you handled the Cs, then move on to the Ds. You have D to the 15th over D to the 8th. Like we just did, apply your rule, and subtract your exponents, right? So 15 minus 8, that's 7. So, this would be D to the 7th. When I simplify those Ds, I get D to the 7th. So, when you have an exponent that's 1, you generally never see it written. You would probably see this as C, D to the 7th. If you did write that 1, you're not incorrect. This simpler way to write it is how you'll see it most often. Okay? Let's keep going. This one, you see, we've got three variables. G to the 15th, H to the 22nd, K to the 9th over G to the 10th, H to the 18th, K to the 6th. We apply our same process and take the variables together, the like ones at the same time. Let's deal with the Gs first. So G to the 15th divided by G to the 10th. We're just going to subtract our exponents. So 15 minus 10, that's 5. I know when I handle the Gs, I end up with G to the 5th. Then, I'd move on to the Hs. I have H to the 22nd over H to the 18th. Just subtract my exponents. All right, 22 minus 18, that's 4. This is just H to the 4th when I simplify that. Last, but not least, let's get the Ks. We have K to the 9th over K to the 6th. Just subtract my exponents. 9 minus 6, that's 3. That part would just break down to K to the 3rd. My last part of this, K to the 3rd. You're all done, handled all three pieces.

(female narrator) G to the 5th, H to the 4th, K to the 3rd. Okay? Not too bad, right? All right, let's keep going. It is now your turn, okay? Press pause, take a few minutes. Work your way through these problems. When you're ready to check your answers, press play again.

(female narrator) Simplify the following expressions. Number 1: Z to the 30th over Z to the 20th. Number 2: M to the 7th times N to the 8th over M to the third times N to the 5th. Number 3: X to the 18th times Y to the 13th times P to the 4th over X to the 12th times Y to the 9th times P squared. All right, you ready? Let's see how you did. C to the 30th divided by Z to the 20th, that's Z to the 10th for that one. Two: M to the 7th, N to the 8th over M to the 3rd, N to the 5th, was M to the 4th, N to the 3rd. All right? Finally: X to the 18th, Y to the 13th, P to the 4th divided by X to the 12th, Y to the 9th, P squared. You should have got X to the 6th, Y to the 4th, P to the second power. To see how I did these, keep with me. How did I get that first one? Okay, just applied my rule, and subtracted my exponents. So 30 minus 20, that's 10. That means that this will simplify to Z to the 10th. On the next one, I had two different variables, but I applied the exact same process. I handled the Ms first. M to the 7th divided by M to the 3rd, you just subtract your exponents. So 7 minus 3, that's 4. That means the M piece of that simplifies to M to the 4th. Got the first bit. All right, then I have to handle the Ns. I have N to the 8th divided by N to the 5th. Just apply your rule. Subtract the exponents. Okay, 8 minus 5, that's 3, so N to the 3rd. Then you've got the last part of that. That's how I got number 2. If you need to see how I did the last one... This is the one with three different variables. Take them one at a time-- the same type at a time. So X to the 18th over X to the 12th. Just subtract your exponents. 18 minus 12, that's 6, so X to the 6th. That was the first part. Then handle the Ys. So Y to the 13th over Y to the 9th. Subtract your exponents, so 13 minus 9. That's 4, right? That's where the Y to the 4th came from. Then we have the Ps. So P to the 4th divided by P squared. Subtract your exponents. 4 minus 2, that's 2, so P to the 2nd power. That's where the last piece came from. That's how I got those three answers. There's another kind of problem that applies this rule. I want you to know how to do this type, too.

(female narrator) Example 5: 40G to the 5th, Y to the 9th over 20G squared, Y. This is still a problem involving the quotient of powers, but you have coefficients out front. Now you also have to handle that. Come off to the side. It's not a very different process. I need to handle that 40 divided by 20 first. 40 divided by 20, that's 2, right? That means the first part of this is 2. Then I have G to the 5th divided by G squared. G to the 5th divided by G squared. Subtract my exponents here. So 5 minus 2, that's 3, so I've got G cubed. Then I've got the Ys to handle here. I've got Y to the 9th divided by Y. Uh-oh! You see, you don't really see an exponent? Remember, when you don't see an exponent, there's an invisible 1 for your exponent, so just throw it in there to help you get through it. I've got Y to the 9th divided by Y to the 1st. Subtract those exponents. So 9 minus 1, that's 8, so Y to the 8th. Ooh! I wrote 18. Let's get rid of that. Y to the 8th. All right? All done with that one. Okay? Let's try another one. This one, I have 12X to the 9th, Y to the 11th divided by 15, X to the 6th, Y to the 7th. So same process, handle the coefficients first. So 12 divided by 15. All right? Don't be alarmed. That won't divide cleanly like the other one. My answer is not a whole number; it's another fraction. We're not afraid of fractions. We'll reduce this fraction. Remember when we were reducing the fractions? Do the same thing. I need a greatest common factor between 12 and 15. It helps to know your multiplication tables. All right, I know 4 times 3 is 12, and I know 5 times 3 is 15. The greatest common factor between those two terms, those two numbers, is 3, right? I can cancel out the 3s. That means this fraction reduces to 4/5ths. The first part of this problem, the part involving the whole numbers, that part equals 4/5ths, okay? I'll keep moving along and apply my quotient of powers rule to handle the rest. So X to the 9th divided by X to the 6th. You remember what we do? Subtract the exponents. So 9 minus 6, that's 3. So X to the 3rd power. I know that's the next bit. Let's handle the Ys. You've got that Y to the 11th divided by Y to the 7th. Apply your quotient of powers rule. Subtract your exponents. So 11 minus 7, that's 4. So Y to the 4th. Sometimes, you won't see the answer written like this when it's a fraction. You should know the different ways this can be written. You may see it written like this-- 4/5ths times X to the third, Y to the 4th. Or sometimes, people like to separate the numerator from the denominator even further, like really show that division. So, you may see this as 4X to the 3rd, Y to the 4th, all over 5. Those each mean the same thing. It's whichever way you want to write it. They mean the same thing, they're both acceptable. Okay? Let's move along. It is your turn, so press pause. Take your time, work through these problems. When you're ready to compare answers with me, press play.

(female narrator) Simplify the following expressions. Number 1: 75A to the 19th, B to the 18th over 50A to the 11th, B to the 10th. Number 2: 30H to the 9th, V to the 11th, T to the 4th over 5H to the 5th, V to the 3rd, T. You ready to see how you did? Let's compare our answers. For the first one, 75A to the 19th, B to the 18th divided by 50A to the 11th, B to the 10th. That was three halves A to the 8th, B to the 8th, or maybe you wrote it as-- maybe you wrote it as this: 3A to the 8th, B to the 8th over 2. Either one would have been fine. All right, the second one, 30H to the 9th, V to the 11th, T to the 4th divided by 5H to the 5th, V to the 3rd, T. You should have got 6H to the 4th, V to the 8th, T to the 3rd. Okay? See how I did these? Then stay with me, I'll work them out. For the first one, I did what we did before. I focused on those coefficients first. I said, "Okay, 75 divided by 50." Those won't divide evenly, okay? Whenever I see numbers 25, 50, 75, 100, I always think quarters to figure out what my factors are. I think like, "75, that's three quarters," right? The factors of 75, 25 times 3. All right? 50, I know I could factor that with 25 times 2, right? When I do that I see my greatest common factor is 25. I cancel out those 25s. So I got the three halves, the 3 over 2. You notice that's an improper fraction. The numerator is larger than the denominator. In elementary school, people really hated that and wanted to make it a mixed number. When you get to Algebra 1, nobody hates it anymore. Leave an improper fraction as an improper fraction, it's fine. It's easier to leave it like that Algebra. I'll leave that as three halves, okay? Now I'll handle the A part. A to the 19th divided by A to the 11th. I subtract my exponents. So 19 minus 11, that's 8. So A to the 8th. And B to the 18th divided by B to the 10th. Subtract my exponents here. 18 minus 10, that's 8, so B to the 8th. You just remember, like I said, sometimes it's written like this, where you're forced to notice everything that's in the numerator. That 3A to the 8th, B to the 8th, with that 2 on the bottom. Both mean the same thing. I like the first way-- that's my habit-- but either one is fine. It's completely up to you. If you see that second one, same exact process: take the coefficients first, and then deal with the rest. So 30 divided by 5, that's 6. That's where my 6 came from in my answer. Then I handled the Hs. So H to the 9th divided by H to the 5th. Use your quotient of powers rule. Subtract the exponents. So 9 minus 5, that's 4. That's where I got the H to the 4th. Then I handled the Vs. So V to the 11th over V to the 3rd. Subtract your exponents. So 11 minus 3, that's 8. That's where the V to the 8th came from. Then the last bit: that T to the 4th divided by T. Remember how, when you don't see an exponent, there's an invisible 1? So let's subtract 4 minus 1. That's 3, so T to the 3rd. That's where the last bit of that came from. All right, guys! I hope you're using your quotient of powers rule to solve your problems, and that using your knowledge of exponents, division, and factors-- it comes together to help you with these kind of problems. Hope to see you soon.

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In this program, students learn to apply the quotient of powers rule for division. This rule allows for subtracting the exponents if the base numbers are the same. Part of the "Welcome to Algebra I" series.

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Runtime: 28 minutes

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Episode 1
31 minutes
Grade Level: 7 - 12
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Episode 2
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Episode 8
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Episode 10
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