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Welcome to Algebra I: Power of a Power

13 minutes

Hey, guys. Welcome to Algebra 1. Today we'll focus on solving problems involving finding power of a power. What you know about exponents and multiplying and patterns will go a long way with these problems. Ready to get started? Let's go. Okay, to get going with these, I first want to establish a pattern for you. It'll help these type of problems make more sense. Knowing what you do about exponents, you know that 4 to the 3rd power is the same as 4 times 4 times 4, if you expanded it out to be a product of its factors. I have a product of three 4s, okay? Applying that knowledge, how about this one? What if I needed to raise 4 squared to the 3rd power? Let's still think about it. Whatever's in our parentheses, that term, we're going to multiply by itself three times. So, 4 squared raised to the 3rd power is the same as 4 squared times 4 squared times 4 squared. I'm just multiplying 4 squared times itself three times. Let's keep breaking this down and see what it simplifies to even more. I know that 4 squared means that I'm multiplying 4 by itself right? That first little bit there, that's the same as 4 times 4. 4 squared is the same as 4 times 4. The middle one is also 4 squared. That's another 4 times 4. Same thing for the last one. That's another 4 squared. That's another 4 times 4, right? With me so far? I took that from 4 squared times 4 squared times 4 squared, and broke it down even further; 4 squared is the same as 4 to the 4th, same here, same here. Let's get rid of those parentheses so we can just see exactly what's going on here. I'm just going to rewrite it without the parentheses. Break those 4s out of the groups. When I see this expanded out, how many 4s am I multiplying by itself? Let's see--I've got one, two, three, four, five, six. That means I can represent this as 4 to the 6th. When I expand it all out, I started at 4 squared raised to the 3rd power. When broken down, it's the same as 4 to the 6th. Let's keep applying that and get through a couple more problems. I'll do a few of them. You'll notice the pattern as I get to a certain point. Same thing, you know what we do about exponents. If I'm raising 3 to the 4th to the 2nd power, that means I'm multiplying 3 to the 4th by itself. I have 3 to the 4th times 3 to the 4th. I'm going to expand this out and really look at it as the product of 3s. 3 to the 4th, that means I'm multiplying 3 by itself four times. That first bit, that's the same as 3 times 3, times 3 times 3. 3 to the 4th, product of four 3s. The second bit is also 3 to the 4th, so 3 times 3 times 3 times 3. I just expanded that out. Now I'll scrap those parentheses and look at these 3s and see how many I've got. I've got one, two, three, four here, multiplied by another one, two, three, four over here. How many 3s am I multiplying together here? Let's count them out. One, two, three, four, five, six, seven, eight. That means I could represent this as 3 to the 8th. So, 3 to the 4th, raised to the 2nd power, is the same thing as 3 to the 8th. Let's keep going. Have you noticed the pattern yet? If not, don't worry about it, you will. This one, I've got X squared, raised to the 5th power. So I'm multiplying X squared by itself five times. So X squared times X squared times X squared times X squared times X squared. I bet you're realizing why someone sought a pattern, because depending on how large those exponents are, you could be writing all day. Let's keep going with this. Each of those X squared means the same thing as X times X. You're just multiplying X by itself. For that first X squared, you've got X times X. Same for the second one. Same for that third one, another X times X. Same for that fourth one, another X times X. Same for that fifth one--ooh! Didn't write it, just copied it. Let's erase. There we go, bring it back. I'll break that down. X times X. And that last one, X times X. Okay, so we wrote it in groups like this. Now let's break all those Xs out of the parentheses and see what we're working with. I've got X times X times X times X times X times X times X times X times X times X. How many Xs are you multiplying together? Let's count them out. One, two, three, four, five, six, seven, eight, nine, ten. You multiplied X by itself ten times. That means you could represent that product as X to the 10th. So X squared raised to the 5th power is the same thing as X to the 10th. Keep with me. Let's look at those three problems and their answers all together. Looking at them all together, you may notice the pattern. When we raised 4 squared to the 3rd power a few problems ago, that answer was 4 to the 6th. When we raised 3 to the 4th, to the 2nd power, that answer was 3 to the 8th. And when we raised X squared to the 5th power, that answer was X to the 10th. Pause me for a minute. Look at that. See if you can figure out the pattern. Do you notice any relationship between the exponents in my problem, and then the exponent in the answer for each of those? Pause me and think about that. Did you see it? Did you figure it out? This is what was going on. If you multiply these exponents together, 2 times 3, that's 6. If you multiply these together, 4 times 2, that's 8. If you multiply these together, 2 times 5, that's 10. That's the pattern with these. Any time you raise a power to another power, the shortcut is to multiply the exponents together. Don't go through all that work like we did-- expanding it all out, finding the product, looking at all those factors. You can just follow the rule for raising a power to a power. The rule is just this: whatever your power is on the inside, whatever you're raising to, you just multiply those exponents together. Just to say it generally, you'll see it like this, maybe with different letters, but A to the R, raised to the S power-- it's the same as A times R to the S. Multiply your exponents together and you'll get your problem all simplified. Let's try a couple. We're just going to use the rule. Don't expand it; use that rule of raising a power to another power. Here, I've got 5 cubed, raised to the 10th power. All I need to do here is just multiply my exponents. Just multiply the exponents. That is the shortcut here. So 3 times 10, that's 30. That means that this simplifies to 5 to the 30th. If you took the time to expand that, and write all those fives, you'd be multiplying 30 fives together. Nobody wants to multiply 30 fives together, so represent it like this: 5 to the 30th. Let's try a couple more. I've got P to the 3rd raised to the 11th power. Remember, just multiply your exponents together. I'll come off to the side. 3 times 11, that's 33. That means that this simplifies to P to the 33rd power. You're all done; that would be your answer right there. P cubed raised to the 11th power is the same as P to the 33rd. All right, let's keep going. All right, so your turn. Look at these, go ahead and pause me. See how you do with these. To compare your answers, press play and we'll see how we did.

(female narrator) Simplify the following expressions. Number 1: 3 to the 9th, all to the 2nd. Number 2: W to the 7th, all to the 5th. Okay, ready to check? Let's see. 3 to the 9th, raised to the 2nd power, that is the same as 3 to the 18th. And W to the 7th, raised to the 5th power, that's the same as W to the 35th. To see how I did those, in case you got something different, here's what I did. So 3 to the 9th, squared, I'm raising a power to another power. So just multiply my exponents together. 9 times 2, that's 18. That means that this simplifies to 3 to the 18th. All done. That next one, I had W to the 7th, raised to the 5th power. So just multiply those exponents together. 7 times 5, that's 35. That means this simplifies to W to the 35th, and you're all done. All right? All right, guys. Hope you've mastered solving problems involving having to raise a power to another power. Apply your rule of multiplying your exponents together, and I'll see you soon for some more Algebra 1. Bye!

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In this program, students learn how to raise one power to another. This rule simply states that the exponents are multiplied. This is true for positive and negative exponents. Part of the "Welcome to Algebra I" series.

Media Details

Runtime: 13 minutes

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