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Pre-Algebra: Quotient Properties of Exponents


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About this Lesson

  • Type: Video Tutorial
  • Length: 3:58
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 42 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 teaches us about the quotient properties of exponents. Exponential properties dictate that a^m/a^n = a^(m-n). Basically, if the bases of two exponential expressions are the same, we can simply subtract the exponent of the denominator from the exponent of the numerator to write the result of dividing one of these expressions by the other. In this lesson, Professor Burger shows you several examples to make sure you are comfortable subtracting exponents in equations that involve exponents, both positive and negative, in fractions.

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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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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Let’s take a look at some examples. X5 Y3 divided by XY3 ––what do we do. Well here what I am going to do is say okay. I see X is here and X is here so this is going to be X5 minus the exponent here which is invisible 1 times Y3-3. And so I see this actually equals X4 Y0 which of course just equals X to the four times one where we understand that Y is not zero because I can’t divide by zero and X is not zero because I can’t divide by zero here. So the answer is just simply X4, that’s pretty fun, looked pretty complicated, didn’t it? It turns out all it is X4 as long as you promise that X and Y neither of them are zero.

What about this crazy, crazy, crazy thing? Well let’s think about this crazy, crazy, crazy, crazy thing. Here I see, I have the common base of X here so I am going to have X2-2. What do you think about that? I hope you don’t like it because it’s actually wrong. I am going to take 2 and subtract off this exponent which itself is a negative two. So I should see 2-(-2). Great classic mistake, be really careful 2-(-2) you see it. You are fooled by that make a note, Y(-3)-3, what does this equal? Well 2-(-2) is the same thing as 2+2 so that’s X4. And Y(-3)-3 is zero, right. Right, right, no. -3(-3) is actually Y to the (-6), be really careful it’s so easy to make easy arithmetic mistakes after doing all this fine work. What does that negative exponent mean? It means that I put it underneath, I take a reciprocal but this has no negative exponent, so this comes to the numerator and this comes to the denominator. And that happens precisely as long as we have X not zero and Y not zero. And so here is but the next one equals, not pretty, it’s so pretty... one more.

Let’s do one more come on where are you going. The quantity (P2Q)4 divided by P3Q2 so first I am going to deal with this exponent on the top here deal with these exponents. Use the properties I have already seen so I can write this as (P2)4 times Q4 well (P2)4 that’s a power to a power so that’s going to be P to the two times four which is eight. Two times four is eight and then I have a Q4. And then in the denominator I have P3Q2. And I can simplify P to the, well subtract the exponents on the like basis 8-3 Q4-2 and so I see P5 times Q2 and of course the rule is P is not zero, and Q is not zero because I can’t divide by zero. And there we have another really, really neat simplified version of this expression under this little condition. Anyway, you can see that when we are dividing two powers that have the same base we just subtract the exponents, exponent in the numerator minus exponent in the denominator, awesome.

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