# Welcome to Algebra I: Product of Powers

19 minutes

(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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