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Welcome to Algebra I: Product of Powers

19 minutes

Hey, guys. Welcome to Algebra I. Today's lesson focuses on the product of powers. You apply what you know about exponents and patterns to work through these problems. Ready? All right, let's go. Okay, to understand product of powers, bear with me for a minute while I take you through problems to set up patterns. It'll help you understand it once we get going. The meaning of 4 to the 3rd-- What does that really represent? My exponent tells me how many times I'm multiplying a number times itself. So, 4 to 3rd power means the same thing as 4 times 4 times 4. It's the product of three 4s, right? Okay. 5 to the 4th. Same idea, right? The exponent tells you how many times to multiply 5 by itself. In this case, because my exponent is 4... I'm going to multiply 5 times 5 times 5 times 5. I've got the product of four 5s. Okay? Now, let's keep going. x to the 5th-- same idea. Your exponent is 5, so I should see the product of 5 "x"s. So, one, two, three, four, five. All right, x to the 5th. You start seeing what's going on with these exponents. What if you have something like 5 squared times 5 to the 4th? What does that really represent? Let's break it down. I know that 5 squared-- that's 5 times 5. Then I'm multiplying that times 5 to the 4th. So, I know 5 to the 4th-- I should see the product of four 5s for that part, so 5 times 5 times 5 times 5. When I look at the problem, I have the product of one, two, three, four, five, six 5s. I could say that 5 squared times 5 to the 4th is the same as 5 to the 6th. When I represent that, expand it all out, look at it all together, I have the product of six 5s, right? Keep going. What about 3 cubed times 3 squared? Use that same idea we did on the last one and see what happens with this one. I know 3 to the 3rd, that's the product of three 3s. 3 squared, that's the product of two 3s. When I look at this, I'll expand it like this. I see that I have the product-- count them out--of five 3s. So 3 to the 3rd--oop-- let's get that pen back. Let's get this out of my way. Let's get that pen back. There we go. 3 to the 3rd times 3 squared is the same as 3 to the 5th. When you look at that product expanded, you have the product of five 3s, okay? All right. Look at this one: x to 3rd times x to the 4th. Getting the hang of what's going on here? I know x to the 3rd-- that's x times x times x, the product of 3 "x"s. I'm multiplying that times x to the 4th, so the product of 4 "x"s-- one, two, three, four. When I look at that product expanded out, I have one, two, three, four, five, six, seven-- I've got the product of 7 "x"s. So x to 3rd times x to the 4th is the same as x to the 7th. Okay, I wonder if you started to notice a pattern. If not, let's try one more. We'll bring it together and see if you can spot it. For this one, I've got 4 times 4 squared. When you don't see an exponent, you can pretend there's an exponent of 1 because that doesn't change anything. That's just one little 4 standing on its own. Then you're multiplying it times 4 squared. That is the product of two 4s. When I look at that product expanded, I see I have the product of three 4s. That means that 4 times 4 squared-- remember, we could think about that as 4 to the 1st-- is the same as 4 to the 3rd, right? We proved that by expanding it all out. Let's look at everything and see if you notice what's going on. These are the problems we did. We're comparing their answers all together on one page. We remember 5 squared times 5 to the 4th gave us 5 to the 6th. 3 square--I'm sorry-- 3 cubed times 3 squared, that was 3 to the 5th. x cubed times x to the 4th-- that was x to the 7th. We said to put a 1 there. Think about it like that-- 4 to the 1st times 4 squared-- that's 4 to the 3rd. Pause the tape if you want, and look at these answers and problems. Look at how they started and see if you can figure out a pattern. Did you notice a relationship between those exponents? Take a look back here. In this problem, my exponent was 2 and 4. Now, 2 plus 4 equals 6. For this one, my exponent was a 3, exponent here was a 2. 3 plus 2, that equals 5. I had exponents of 3 and of 4. 3 plus 4, that's 7. Exponent of 1, exponent of 2-- my exponent for my answer was 3. The pattern with these kind of problems, if you're multiplying two terms that have the same base, like that same-- Here it's 5, here it's 3. The base is the same. Add your exponents together to get the answer. That is how we came about the rule which is called "product of powers." When multiplying two terms with the same base and they have exponents, to get the answer, add the exponents. You don't have to go through and expand that problem and write out those factors, because you know a shortcut-- just use the rule. Add the exponents together when multiplying and your bases are the same. Look at this one. We'll use the rule. Make sure I got my pen. I got y to the 8th times y to the 4th. My bases are the same. They're both y. I simplify this by adding my exponents. I'll come off to the side. You're adding your exponents. Okay, so 8 plus 4, that's 12. That means that y to the 8th times y to the 4th, that's just y to the 12th. You just add your exponents together, and you're all done. You simplify that, okay? Let's keep applying this rule. Keep going. 7 to the 10th times 7 to the 15th. Same exact thing. We add those exponents together because our bases are the same. I'll come off to the side-- 10 plus 15, that's 25. That's what I get when I add those exponents. That means the answer is 7 to the 25th, and you're all done. That's it. You simplify that also, okay? Let's keep moving. It's your turn now. Pause the tape. Take a few minutes. Work through these problems, and we'll get together and compare our answers.

(female describer) 1. x to the 11th power times x to the 5th power. 2. 5 to the 8th power times 5 to the 20th power.

(teacher) Okay, took some time. Let's compare our answers. The first one, x to the 11th times x to the 5th-- Let's get rid of the pen so I could reveal this. That's x to the 16th. 5 to the 8th times 5 to the 20th. That's 5 to the 28th. To see how I did these-- maybe you missed it or you want to check your work. I'll work those out for you. Get that pen back. This one, x to the 11th times x to the 5th, just need to add those exponents together-- 11 plus 5, that's 16. That means the answer to this: x to the 16th. That's how I got that one-- add those exponents together. On the second one, I had 5 to the 8th times 5 to the 20th. Just add your exponents together-- 8 plus 20, that's 28. That means my answer here is 5 to the 28th, and you're all done. Okay. That's how I did those two.

(female describer) Example 3: Simplify 6g cubed, h to the 4th times 2g to the 10th, h to the 5th. All right, we're going to apply the same rule here. This problem is a little different. If you notice, we are still multiplying two terms together, but our terms have a coefficient. I have a coefficient of 6 right here and a coefficient of 2 here. I'm still just finding the product of powers. What I do here, just to help us understand, I'm going to rearrange this multiplication. The commutative property tells us we multiply in any order. It doesn't change the problem. I bring my coefficients out front so I can look at those together. Then I'm going to group my like exponents-- not the exponents-- but the variables together. It helps me organize this and break it down further. I'm going to put that 6 times 2 out front. Get those coefficients together. Now I'm going to multiply g to the 3rd times g to the 10th and get those two terms together, so I handled those. I get that h to the 4th times that h to the 5th and get those together. Okay. Now I look at everything together. I keep working my way through this problem. Now, the associative property says I can group when I'm multiplying. It doesn't change anything. I group those like terms together and handle them in their individual groups. Right here-- 6 times 2, I know that's 12. For g to the 3rd times g to the 10th, I apply the rule, that product of powers. I add exponents to get the answer for that one-- 3 plus 10, that's 13. I have h to the 4th times h to the 5th. Just add those exponents together: 4 plus 5, that's 9. So, h to the 9th. You're all done. That one is the same idea. They threw a curve ball because you got coefficients. We group the like things together and use rules you already know. Okay, let's try another one: 8t to the 3rd, j to the 10th times 5tj to the 7th. Remember what we just did. We're going to rearrange this at first. I'm going to pull the coefficients out to the front. I'll have 8 times 5 out front. Then I'm going to take those "t"s, put them together. So, t to the 3rd times t. I'll get those "j"s and put those together. So j to the 10th times j to the 7th. Okay. I'm going to use my parentheses to group them off. I'm going to handle each group, each chunk separately. So, 8 times 5, I know that's 40. Then t to 3rd times t-- when you don't see an exponent, there's like an invisible 1 there. I need to add those exponents together-- 3 plus 1, that's 4. So, t to the 4th. I have j to the 10th times j to the 7th. Add my exponents together: 10 plus 7, that's 17. You're all done. That expression just simplifies to 40t to the 4th, j to the 17th. Just applying those rules that we learned. Now it's your turn. Try these two problems. These involve the coefficients we covered. Pause the tape, then play it when you're ready to compare answers.

(female describer) Simplify the following expressions. 1. 9m to the 6th, n to the 7th times 5m to the 4th, n squared. 2. 10x cubed, y to the 5th times 8x squared, y to the 6th.

(teacher) Okay, let's see how you did. Get the pointer tool. That first one, 9m to the 6th, n to the 7th, times 5m to the 4th, n squared-- should have got 45m to the 10th, n to the 9th. The second one, 10x cubed, y to the 5th times 8x squared, y to the 6th-- that answer was 80 to the 5th, y to the 11th. If you need to see how I did either one, then stay with me. Let's get that pen and work this out. Remember, our process that we're following. Bring coefficients out front, then group the like terms together. I'd have 9 times 5 for my coefficients. Then I have m to the 6th times m to the 4th. Then I have n to the 7th times n squared. Okay? Now I throw my parentheses in there to group them off and handle them in chunks. So, 9 times 5, that's 45. m to the 6th times m to the 4th-- we're just going to add those exponents together. So, 6 plus 4, that's 10. I have m to the 10th. Then I have n to the 7th times n squared. I'm just adding those two exponents together-- 7 plus 2, that's 9. That's how I got that answer: 45m to the 10th, n to the 9th. You see the next one? Keep going. I have 10x cubed, y to the 5th times 8x squared, y to the 6th, so same process. Get those coefficients together. 10 times 8. Then I have x cubed times x squared. I've got y to the 5th times y to the 6th. You remember what we did next? Threw those parentheses in, group them off. Okay, 10 times 8, that's 80. I've got x cubed times x squared. So just add those exponents together: 3 plus 2, that's 5. I've got y to the 5th times y to the 6th. Just add those two exponents together: 5 plus 6, that's 11, okay? That's how we got that one. All right. Well, you have completed your lesson on product of powers. I hope your knowledge of patterns and exponents helped you through these problems. I hope to see you here soon. Bye.

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In this program, students learn the exponent product rule. This applies to problems that have the same bases. If the bases are the same, then the rule allows for the shortcut of adding the exponents. Part of the "Welcome to Algebra I" series.

Media Details

Runtime: 19 minutes

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Episode 1
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Grade Level: 7 - 12
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Episode 2
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