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Pre-Algebra: Simplifying with Negative Exponents


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About this Lesson

  • Type: Video Tutorial
  • Length: 3:21
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 36 MB
  • Posted: 01/28/2009

This lesson is part of the following series:

Pre-Algebra Review (31 lessons, $61.38)

In this lesson, Professor Burger explains what to do when you have a negative exponent. When there is a negative exponent, the expression will need to be set up as a reciprocal in order to evaluate and simplify the expression. A term with a negative exponent is equal to the reciprocal of the same term with a positive exponent. Thus, x^(-2) = 1/(x^2). Professor Burger explains examples of how to do this with negative integers, subtracting terms, and using variables.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course, Pre Algebra. This course and others are available from Thinkwell, Inc. The full course can be found at The full course covers whole numbers, integers, fractions and decimals, variables, expressions, equations and a variety of other pre algebra 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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Let’s take a look at some quantities that actually involve negative exponents. Here’s the first one. Just for fun let’s throw it out there. 3 to the negative 2 power. Well, what do we do here? Well, that negative in the exponent tells me now you take a reciprocal, and so this actually equals 1 over (3 squared). Notice how that negative 2 became a 2 that took the reciprocal, this equals 1 9th. How about this one, quantity negative 2 raised to the negative 5 exponent? Well, notice that I have parentheses around the negative 2 that means that I need to take the quantity negative 2 and raise it to the negative 5. What does that equal? Well, I take a reciprocal and so I see one over negative 2 to the 5th. What’s that? Well, if I take negative 2 and multiply it by itself five times, 5 is odd, so the answer is going to be negative. And what’s 2 to the 5th, it’s 32 and so I see 1 over negative 32. But we’d never like to have negatives, downstairs in the denominator so I am going to write this as negative 1, 30 second, much cleaner, much neater, and we are good to go. Let’s see if we can evaluate this crazy expression. Now, here the negative 2 is not lassoed with a parentheses. So remembering orders of operation, I first have to do the exponent and look at 2, only 2, that’s the base raised to the negative 4, take that answer and put a negative sign in front of it. So what I see here is negative 1 over 2 to the 4th minus and then what’s this quantity, 16 to the negative 1, that means I am looking at 1 over 16 to the 1 which is just 16. What’s 2 to the 4th? 2 to the 4th is actually 2 times 2 which is 4, times 2 is 8, times 2 is 16. And so in fact, this equals negative 1 16th minus 1 16th which equals negative 2 16ths which is just negative 1 8th. So it simplifies down to negative 1 8th. And finally, for your mathematical pleasure, let’s put some ideas together and compute this. What do we see here? Well, I have this quantity raised to the negative 3, that means I have 1 over that quantity cubed. And then the laws of exponents tell me since I am multiplying here and raising to the 3rd power, this equals 1 over X cubed multiplied by (Y squared) cubed, and I can actually rewrite this a little bit more as 1 over X cubed and I have got a power raised to a power. So I multiply and I see Y to the 2 times 3 which is 6. And so I see the answer is 1 over X cubed Y to the 6th. And so you can see this notion of what it means to have a negative exponent. It basically just means take a reciprocal, take a reciprocal. Anyway, great work, hope you’ve enjoyed negative exponents as much as I do. It just means we are introducing fractions, that’s all. See you soon.

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