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Beg Algebra: Applying the Rules of Exponents


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About this Lesson

  • Type: Video Tutorial
  • Length: 10:11
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 110 MB
  • Posted: 12/02/2008

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra: Basics & Prerequisites (37 lessons, $52.47)
Beginning Algebra Review (19 lessons, $37.62)
College Algebra: Exponents (4 lessons, $4.95)

Professor Burger introduces you to the rules of exponents, including a classic mistake made when multiplying two numbers of the same base with different exponents. You will learn that A^n * A^m = A^(n+m). Then Professor Burger will teach you the next rule, what to do when you multiply two numbers of different bases, raised to the same power (A^n * B^n = (AB)^n). The final rule of exponents teaches you what to do when you have a base raised to an exponent, with the entire expression raised to another power (or (A^n)^m).

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Beginning Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers linear equations, inequalities, polynomials, rational expressions, relations and functions, roots and radicals, quadratic equations and systems of equations.

Edward Burger, Professor of Mathematics at Williams College, earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.

He has also taught at UT-Austin and the University of Colorado at Boulder, and he served as a fellow at the University of Waterloo in Canada and at Macquarie University in Australia. Prof. Burger has won many awards, including the 2001 Haimo Award for Distinguished Teaching of Mathematics, the 2004 Chauvenet Prize, and the 2006 Lester R. Ford Award, all from the Mathematical Association of America. In 2006, Reader's Digest named him in the "100 Best of America".

Prof. Burger is the author of over 50 articles, videos, and books, including the trade book, Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas and of the textbook The Heart of Mathematics: An Invitation to Effective Thinking. He also speaks frequently to professional and public audiences, referees professional journals, and publishes articles in leading math journals, including The Journal of Number Theory and American Mathematical Monthly. His areas of specialty include number theory, Diophantine approximation, p-adic analysis, the geometry of numbers, and the theory of continued fractions.

Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.

About this Author

2174 lessons

Founded in 1997, Thinkwell has succeeded in creating "next-generation" textbooks that help students learn and teachers teach. Capitalizing on the power of new technology, Thinkwell products prepare students more effectively for their coursework than any printed textbook can. Thinkwell has assembled a group of talented industry professionals who have shaped the company into the leading provider of technology-based textbooks. For more information about Thinkwell, please visit or visit Thinkwell's Video Lesson Store at

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Page 1 of 2 © Thinkwell Corp.
Okay, so let’s talk about the rules of the road when it comes to exponentiation. The bottom line is as long as you take it slowly, take it carefully, you will never go wrong. But it’s when sometimes students just don’t stop to think about things that all of a sudden things get a little bit harried. So, let me just go through and show you not only the rules of the road, but how to always get the rules of the road correct.
So, let’s first of all start off with a very simple kind of arithmetic question. Suppose I wanted to figure out what 2-cubed times 2 is. Okay, well, how would I do that? Well, what do you do with those exponents? Do you add them? Do you subtract them? Do you multiply and divide them? A great guess, by the way. Here’s a super guess, this [points to the common base] equals 2 and since I’m multiplying, I get 15. That is a great guess. I hope that you, everyone you know, makes that guess right now. And then never makes that guess again. Because, in fact, this is a classic, classic mistake. In fact, it is number six on my list of classic mistakes people have made. And the mistake is to multiply the exponents when you are multiplying the numbers. That is actually incorrect. And my little mantra to you is very simple: when in doubt, write it out. And this, really, is going to be handy. Let me show you exactly how handy this will be. to the fifth
Let’s write this out. If you were to write this out what would you see? Well, you would see a heck of a lot of twos. There’s 2 times 2 times 2, and then I’ve got 2 times 2 times 2 times 2 times 2: that’s the answer. Now, can you compactify that? You sure can because this equals 2. So, the answer is this equals 2, not the great guess of 2. Well, how do we get 2 to the eighth? Well, in fact, just by writing it out you can see exactly what is going on here; there’s no mystery. What’s happening? Well, I’ve got three twos from the first number and I’ve got five twos from the next number. So, how many twos do I have together? What do I do? I add the exponents; it’s as simple as that. Because I’ve got so many twos here and so many twos there, so in total I have eight of them. So, when multiplying out the bases like this, the secret is to add the exponents, not multiply. So, in general, let me actually give you a general rule of the road here. If you take, for example, A, and you raise it to the n power, and you multiply that by A, then what that equals is A power. There is a rule of the road. So, that is a great fact about exponentiation that we’ll use again and again and again. to the eighth to the eighth to the fifteenth mn plus m
Now, what about when you have different bases? When, in fact, you just don’t have A and A together? Let me show you an example. For instance, suppose that I have 3 and I multiply that by 5. What would be a great wrong answer here? Well, a great wrong answer would be to say, “Add exponents.” Now the bases are different—I’ll just multiply the bases together—so that would be 15—add exponents—trying to not break cardinal rule number six—and to the fourth to the fourth 4 plus 4 equals 8. So, there’s a great wrong answer [15 ]. Why is it wrong? Well, again, all you have to do when in doubt is write it out. If you write out what this thing means exactly, you’ll see why this is wrong and what the correct answer is. Let’s write it out: 3 to the eighthtimes 3 times 3 times 3 and then 5 times 5 times 5 times 5.
And by the way, on a quiz and stuff if you are not sure about the rules, or you forget the rules and are a little bit insecure, do it. Maybe not with such big exponents, but just do it with small exponents and you will always find what the correct formula is. Because look what I can do here: I have a 3 and a 5 that match up perfectly, and then I’ve got another 3 and a 5 that match up perfectly, another 3 and a 5 that match up perfectly, and another 3 [points to 3 and 5]. They match up perfectly. I can put them together, and I could write it like this, because with multiplication the order doesn’t make a difference. 3 times 5, 3 times 5, 3 times 5 and that last special one, 3 times 5. You see I put them together: (3 times 5) (3 times 5) (3 times 5) (3 times 5). So, what do I see? This is 15, 15, 15, 15, and how many do I have? I have four of them. So, this actually equals 15. So, this [15] was actually wrong. When you have different bases, but you are raising them to the exact same power, then, in fact, you can combine them like this and just take the product of the bases raised to the same power. Is that a great mystery? No, not at all, it’s just a matter of writing it out and seeing it. to the fourth to the eighth
So, in fact, let me write that rule out. That rule would look like this: I have A and a different base B, but to the same power, B, this is just AB all to the n, (AB). That’s pretty easy and that’s not a problem at all. nnn
Okay, let’s see, how about if we wanted to look at something really complicated? By the way, division works the same way; division and multiplication are just twins. So, let me just write that down real fast. If I have A to the n divided by B to the n, but to the same power, then this is just A divided by B to the n. I always forget to mention division: it is sort of sad because it is sort of getting a bum rap on this. Division is good, but it is always the same thing as multiplication in these cases with exponents. So, okay, great that takes care of multiplying two things where the bases are the same. You add
Page 2 of 2 © Thinkwell Corp.
the exponents. If in fact the exponents are the same, you keep the exponent and just multiply the bases. And where did this come from? Just by writing it out, and you can see all the terms there.
Let’s try now one example using these rules, and let’s consider the following. Here’s the expression (2xy). Let’s see if we can write that out without any parentheses at all. So, what’s the first thing that I would do? I would see that I have three terms here and I’m cubing the whole thing. That means that I would first cube this term, I then cube this term, and I then cube that term. If I were to write this out in great detail—more detail probably than you’d want to see—it would be (2) )(x) ) (y) ). Now, how are we going to deal with this? Well, 2 you know is 8, so that’s not a problem. But what about this term right in here? What about (x)? Well, how would you deal with that? Well, again, the way you would think about how that would work is to write it out. to the fifth to the fourth -cubed-cubed to the fifth-cubed to the fourth-cubed-cubed to the fifth-cubed
Let’s do a real simple example and then come back and see why this would be. Let’s put in here, for example, (2). Write it out. So, what do I see? I see 2, 2, and 2. What do I do when the bases are the same? I add the exponents. And so what I would see here would be 2 to the 5 plus 5 plus 5 equals 15. What’s the rule? How, with five and three, do I get 15? I multiply. So, in fact, when I see something to a power and that whole thing is raised up to another power, I multiply the powers together. And that’s the rule. In fact, let me write that down as an actual rule so you can have that. 53 to the fifth to the fifth to the fifth
In general, what that says is if you take A that’s raised to the n, but then you raise the whole thing to the m power, what you have is A to the . A great mistake, by the way—let me show you what a really great mistake is—this would make my top ten list. You want to know all the mistakes, don’t you? Here’s a great mistake. This one is a real winner. You take A to the n, raise it to the m, and then you say that’s going to be A to the n raised to the m power. Now that’s a biggie. That’s a biggie, but it’s not a correctie. It’s a wrongie but a biggie. This is wrong. And the actual reality is this—and again, it’s not something you should memorize, it is just something that you should understand—and the idea is that if you just write it out, you’ll see that you’ll have m many n’s. If you put it them all together, it’s mnm times n.
Armed with that, we can now proceed, because what happens here? Well, this is just a friendly eight. Here I see x to the—now is it 5, is it 5 plus 3, is it 5-cubed times 3? Well, it’s multiplication because I’m raising something to a power to another power. So I multiplied these and I get 15. That’s what we just saw. And then I have y to the 4 times 3, which is 12. So, in fact, this expression can be written without any parentheses just as 8xy. And there you have a way of using these tools and facts about laws of exponents to actually simplify things. And we’ll use that an awful lot. to the fifteenth to the twelfth

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