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College Algebra: Expression with Negative Exponent

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About this Lesson

  • Type: Video Tutorial
  • Length: 9:33
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 103 MB
  • Posted: 06/26/2009

This lesson is part of the following series:

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

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, College Algebra. This course and others are available from Thinkwell, Inc. The full course can be found athttp://www.thinkwell.com/student/product/collegealgebra. The full course covers equations and inequalities, relations and functions, polynomial and rational functions, exponential and logarithmic functions, systems of equations, conic sections and a variety of other AP algebra, advanced algebra and Algebra II topics.

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.

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Okay, so now in our road through all the rules of exponentiation there is this issue of, okay, we know how to put things together we are multiplying and stuff. What about when you are dividing? So, we talked a little about division of some things, but now I want to talk a little bit more about that. Again, remember with exponentiation you can always go back to the fundamentals and just write it all out. Let me try to illustrate that with an example.
So, suppose that we want to take a look at the value of 2^5 2^3. Now, how could you make progress with this? Well, okay, there is going to be canceling that we can do, but the way to see this easiest is just to write it all out. I'm not proposing, by the way, that you do this in every single problem. I'm proposing that you do this whenever you are not quite sure what the right rule is, because this will always work. So, they write out three copies of the 2 on the denominator and five copies of the 2 out on the numerator, then it is easy to see what I can do. I can cancel all sorts of good stuff. Like, this two can cancel away with that 2, this 2 cancel with that 2, this 2 cancel that 2. And, on the bottom, remember, we don't have a zero, but we have our invisible friend Mr. One, so we have a one down there, on the top we have 2 x 2 which is 4. So, this whole thing equals 2^2 or just 4.
So, how did we get from the 5 exponent and the 3 exponent in the denominator to a 2 exponent? And the answer is we subtracted it. And, you can see that here, right, there's no mystery here. I've got 5 on top and I subtracted off the 3 I had on the bottom and that leaves me with 2. So, in fact, there is a fundamental rule here that is really easy. That is the following, if I have A^n and I divide it by A^m, then what am I left with? I'm left with A^n-m. Notice that I take n minus m, the exponent on the numerator minus the exponent on the denominator. A really great little mistake would be to take m minus n, but if you think about what that means and think about the cancellation, you will see that this is the right way to go.
Okay, fun. Well, in fact, this is really fun, because now I can tell you all sorts of neat stuff and show it to you, not just tell you it's like show and tell. For example, let's suppose that I am thinking of any number at all, A for any number. Now, let's just pretend that it's not zero, because, of course, dividing by a zero, I'll put a zero down there, that ain't too good. Let's take an A that's not zero, it could be negative it could be positive I don't even want you to tell me what it is. Now, look what happens? If I take A and divide it by itself, well, on the one hand, if I take A and divide it by A, well, I know what I get. You cancel and you get 1. "Why am I wasting your time?" you're asking. Well, let's now use this fact, this is A^1 and this is A^1 and now I see how to combine them. I subtract the exponents. And, what do I get? I get A^-1, zero.
So, we just discovered a really neat fact. If you take any nonzero number and raise to the zero power it always equals the number one. Isn't that cool? Now, people hear that thing and they always go, "Why is that true?" Well, now you see, it's just the fact about canceling away. You can see it for yourself. So, anything to the zero is one. Now, notice we can't put zero to the zero mathematicians start to shake and they get the willies, because that would sort of mean that we are dividing by zero and as you know that is not good. So, we don't do that, but if A is any number that is not zero, you raise it to the zero power you always get one. So that is a great consequence of this fact.
I want to now show you another great concept. This is a fact that has so many little nooks and crannies you can just have fun with it all day. It is sort of like bringing out a party because people will love playing with it. Let me write it back up here really fast. If I have A^n and I divide it by A^m then this equals A^n-m. Now, with this fact we can do something else. Suppose that I look at just this nothing on top, well, what does that equal? How could I write that as just A^a? Well, the way to do that is to use what we already figured out. I can write one as A, namely . Notice how I am just not doing anything new, I'm just doing variations on what we have already discovered, that's all that math is. And, what's the recipe? You subtract, but remember the order; it's the numerator exponent minus the denominator exponent. So, that is going to be A to what power? Zero minus m, well that's just minus m. So, now, through this neat formula that you all ready are convinced is worth its weight in gold, we get another freebie. We get is the same thing as A^-m. That is to say we finally understand or I finally understand what a negative exponent would mean. We all know what A^5 means it's A A A A A. But, what would A^-5 be, what does it mean to take A and multiply it by itself negative five times? Ain't going to happen. But, now we see what that should mean. It should mean the reciprocal of A^5. So, A^-5 would be . So, that's really cool when you see negative exponents that means you take a reciprocal. And, so that is a really neat fact, which all came from this.
Now, let me just look at some real quick examples to drive these ideas home. The first one let's take -3 that whole quantity and raise it to the -2 power. Now, what would that be? Well, let's think about that. First of all, the way I tackle these problems by the way is to look at the exponents and very slowly untangle them. Again, always trying to take a hard problem and break it into the basic components and move from there. So, let's see that negative exponent always gives me a little bit of a concern so I want to take care of that immediately. A negative exponent means I take a reciprocal, and so the reciprocal I would take is the following. I would take the whole thing one over, just copy everything down, but now make that a positive 2. Here is a wonderful mistake. Some person might say, let me start with even a better mistake. That was a mistake; that was my mistake that wasn't a wonderful mistake that was a stupid mistake. But now I'm going to show you a really wonderful mistake. A wonderful mistake would be someone thinks, okay, a negative that means I flip it, and so they say something like this, that's a grave mistake. Right? Because they remembered if you have a negative exponent you are supposed to flip something. Of course, that person wasn't thinking about exactly what they are supposed to be flipping, they just flipped something instinctively. If that person would have written the thing out and remembered how to subtract the exponents when you are dividing that person would have been led to realize that you flip this thing and then you put it over with a positive 2.
Another wonderful mistake, by the way, would be to put one over negative three and then still keep the negative two there. There is a person that flipped, nut was so enthusiastic about flipping he kept the negative sign in there, which means he has to flip it again and then that would be the wrong answer. So, remember, when you flip, the negative sign would go away. Okay, well now we are home free because now we just have to square that little thing, which is easy. x -3 is just 9. So, pretty easy.
Let's try another one. How about -3^-3. What would that be? Well, the negative sign here means that I flip the whole thing. And, now, I have to multiply negative three by itself three times and that is going to give me -3 x 3, which is 9, times -3, which is -27. So, it's -27? I like to write answers as -1, the negative on top of the other, but either one is fine, they are correct. So, the answer is , pretty straightforward.
Let's try one last one here. The last one is the following, let's take 3/4 and raise that whole thing to the -2 power. What is going to happen here? Well, first we flip, that would be , then I can square 3/4 and remember the rules of exponents. You just can square each of the terms so 3^2 is 9, 4^2 is 16. Now, what do you do when you have this compound fraction? Remember, this is a complex fraction. You take the bottom and what do you do? Flip it and multiply it. By now we are just multiplying one so this is actually a walk in the park. You just take the reciprocal it would be remember that fact when you got down there, flip, the negative sign gets you down there.
So, negative exponents, not a problem it just means you have to flip the whole thing to the exponent with now a positive number there. That's all.
Prerequisites
Exponents
Evaluating Expressions with Negative Exponents Page [2 of 2]

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