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College Algebra: Evaluate Exponential Expressions


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About this Lesson

  • Type: Video Tutorial
  • Length: 4:20
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 47 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 at 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.

About this Author

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Okay, let's just talk about some really nuts and bolts basics, down to brass tacks little facts about exponentiating that could come in handy latter on in life. So, the first thing of course is what does it even mean? Well, of course, if I had something like you know some number a and I raised it to the fourth power, what that means is that I just multiply a out by itself again and again until I do it four times. So that's all exponentiation means there. It's pretty simple. But you could do of course anything. Don't think you're restricted to a. Ha, Ha, no way, you kidding?
For example, you could take like a duck, all right, and you can square a duck. Now you probably said I haven't squared a duck recently. But, if you were to square a duck, what would you get? You would get duck, duck, goose. Make your own joke folks. Okay, so the point is that just exponentiating you just repeat that thing that number of times. So pretty straight forward.
Okay, let me show some little gray pitfalls that people sometime peer into. Ah, the first one is, for example, if I just say something really ridiculous, like I say 3^2. Okay, well, the whole world would respond with 9. Okay fine, but what if I said minus 3^2? Well, now the world might not be responding in perfect harmony, because they might be saying a whole bunch of different things. Some people may say still 9 because they're thinking -3 x -3 would be +9. Remembering correctly that a negative times a negative is a positive.
But there might be other people that are saying wait a minute. Well, you know 3^2 that's still 9. I got a negative sign out in front. So it's -9. Ha, Ha, so is the answer 9 or -9? If you're a politician you might say it's both. It's plus or minus 9. You know, okay. Well, one of these groups is correct. And of course it's not the politicians. No surprise there. Oh, I hope we can't be sued, but oh well. On the web freedom of speech I can say anything, but I won't.
Okay, how do you look at this? The reality is, is that this is different from something else. Let me show you the other thing, and then maybe you can guess why I'm telling you it's different. What the answer is. See this is a little bit different. This is -3 quantity squared. So, half the population that was saying -3 x -3, negative times a negative is a positive, they were actually doing this problem. This indeed equals 9. That negative sign is captured in there.
Those people that thought that this answer is actually 9 made a classic mistake. And for those people well, my hat is off to them, because they made a classic mistake that actually made my top ten list. This in fact is number ten, just snuck in there. But anyway this is the classic squaring a negative mistake and the idea is saying -3 squared just like that equals 9. So, always remember a minus a squared, unless it's been snared.
Any way the point is that the negative sign has to be in the parenthesis with the exponent on the outside in order for that negative sign to actually be multiplied by itself. So how come this turns out to equal -9? That is, first do the 3^2 and stick on the negative sign.
Well, actually I can explain that to you in a way that will make a little sense. In fact this is an invisible coded message. Let me decode it for you. What this is really saying is the following. We have a -1 multiplied by 3^2. That's what this is actually saying. So, again there is that invisible one, that invisible friend, that I have and it's a -1 that's multiplying this. Well, now you can see it; it's very clear, right? This is going to be a 9 and this is going to be a negative sign so we see -9. And the thing that is very important is that order matters. In fact, what we have is that exponentiation always happens before you multiply. So, the order of how you do algebraic steps, for example if you are using a graphing calculator or even a non-graphing calculator, the computer that is reasonably smart will actually know the right order to do things. And the order is to always do exponentiation first and then do any multiplication second.
So, in fact one thing to think about is that exponentiation or exponents always beat up multiplication, always beats up multiples. So, it's real easy to see that in fact there is a difference between this kind of thing verses this kind of thing. So, that's a good thing to know. Okay, well, that's all I've got to say about the very basics about exponentiation. And if you want to know more, keep searching.
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