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College Algebra: Simplifying Radical Expressions

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About this Lesson

  • Type: Video Tutorial
  • Length: 3:22
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 36 MB
  • Posted: 06/27/2009

This lesson is part of the following series:

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

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 that we have this idea of some of the fractional exponents being as roots. Let's look at some real fast examples together. First of all, how about just a lot of fractions there, a fraction here a fraction here how are we going to deal with this? Always start off by just chipping off a little bit of the problem one at a time. That negative sign in the exponent I'm going to deal with. Remember what negatives do, they flip the whole thing, but flipping a fraction is just reverse it, invert it. So, what I see here, I can get rid of that negative exponent by flipping the quantity , so I got rid of the negative sign by flipping. Okay, now I have to deal with this. What does that mean? Well, this means I have to cube everything and then take the square root. Now you can actually cube this and take the square root or you can use my method, which is to first take the square root and then cube it. I like that better because it actually simplifies things. So I take the square root and I can take the square root of the top and then the square root of the bottom so ala carte, separately, and then cube the whole thing. Well what's the ? The is just 10. What's the that's 11 and now I have to cube all that. So if you cube that I see 1 with a whole bunch of zeroes after it, three of them. And then what's 11^3, well it's 11^2 x 11 which is 121 x 11 = 1,331. So that really awful looking fraction to fraction negative thing turns out to be just , neat.
Let's try one last one so you really can get a sense how this thing works. Let's take the cube root of some big old thing. Let's take what in the world would that equal? Well, the first thing I'm going to do is think about raising this whole thing to the one-third power. Since that power is the same we can remember one of the properties of exponentials; I can just take each of these things individually to that power. So, in fact, this would equal. Now, what do I do? Well, I just do each of these separately. What is the cubed root of 64? Right, 64 means that I have to find the cube root of some number so when multiplied by itself three times gives me 64. And that turns out to just 4, right, 4 x 4 = 16, 16 x 4 = 64. Now what's this? Well, you could take 3^6 if you want and then take the cube root, but if you are lazy, remember the good mathematician is the lazy mathematician, I'll just write it like that and the same thing here, 5 and then I can see this can simplify dramatically. This is 4 and this is 3, 3 to what power? Well, is just 2, and this is 5^1 so in fact there is a lot of cancellation when you use these fractions things for roots. So, what do you see? What you see is 4 x 9 x 5, so that's 45 x 4 =180. This awful looking thing turns out just to be 180.
Notice the power of converting everything to exponents using the properties of exponents carefully and actually producing a nice clean answer for this.
Prerequisites
Radical Expressions
Simplifying Radical Expressions Page [1 of 1]

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