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College Algebra: Rationalizing Denominators

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About this Lesson

  • Type: Video Tutorial
  • Length: 12:22
  • Media: Video/mp4
  • Use: Watch Online & Download
  • Access Period: Unrestricted
  • Download: MP4 (iPod compatible)
  • Size: 133 MB
  • Posted: 11/18/2008

This lesson is part of the following series:

College Algebra: Full Course (258 lessons, $198.00)
College Algebra Review (30 lessons, $59.40)
College Algebra: Basics & Prerequisites (37 lessons, $52.47)

In this lesson, you will learn how to get square roots (or cube or other roots for that matter) out of the denominator of a fraction without changing the mathematical value of the fraction. This process is called 'Rationalizing' the denominator. Mathematicians agree that proper fractions should not have radical signs in the denominator, so when you end up with an answer that has a radical in the denominator, it is best to rationalize (as your answer will be marked wrong or will not appear as a choice if you do not).

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 http://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.

About this Author

Thinkwell
Thinkwell
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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 www.thinkwell.com or visit Thinkwell's Video Lesson Store at http://thinkwell.mindbites.com/.

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Recent Reviews

Nachan_homepage
So easy to rationalize denominators!
01/15/2009
~ nachan

This is a great lesson because professor Burger talks about getting rid of square roots both on top of fractions and below fractions. He then goes on to explain how to get rid of each square root to rationalize the denominator in a step by step, easy to understand way. He explains how to clear a denominator for the radical in one easy step!

Nachan_homepage
So easy to rationalize denominators!
01/15/2009
~ nachan

This is a great lesson because professor Burger talks about getting rid of square roots both on top of fractions and below fractions. He then goes on to explain how to get rid of each square root to rationalize the denominator in a step by step, easy to understand way. He explains how to clear a denominator for the radical in one easy step!

You know, I don’t know really how to start this because this is sort of the dark side of mathematics especially the mathematical community. I don’t mind talking about discrimination and I certainly don’t ever want to imply that the math community discriminates against anybody or anything. But, the reality is it does and I want to tell you about it now. So, it pains me a little bit, but stay the course and I think we can get by this dark moment.
You see if you look down here, here is a very friendly looking number and if a mathematician would see this you know what? That person would have no trouble with this number. They would just go on their merry way. But the sad fact is that if we just reverse the roles and wrote this many mathematicians would be very uneasy and would want to change that. Now why is there so much hatred toward having a square root in the denominator? Why do we discriminate the numerator from the denominator? Well, there are a variety of reasons and some of them actually may be reasonable as we’ll see through the course of this stuff, but the reality is mathematicians often do not like having a square root down below, on top—fine, down below—no. So how do we alleviate this? How do we actually try to make these closed minded people happier? Well, the reality is we can get rid of square roots in the denominator. And, the way we do that it is by remembering that really a square root multiplied by itself is just going to be an unsquare root.
So, for example, here let’s see how we could actually make someone happy that doesn’t like a denominator with a square root. Well, all I would do is, of course one way to make them happy is to remove the whole thing then everyone is happy because there is nothing there, blank—happy. But let’s suppose that we write it back. If we wrote it back that would be . Now there is a great fact in all of life that you should always remember, and that is if you multiply by one, nothing happens. It is fantastic and sometimes doing nothing is more powerful then doing something. In this case it is very powerful, because what I am going to do is I’m going to multiply by one. Now I’m not going to write a one there, like this, that would be sort of foolish to write a one there. What I’m going to do is multiply by a very fancy version of one, namely some number divided by itself, which is equal to one. And that number is going to be exactly the square root of three. So this is not going to change the value of the fraction. It is still numerically equal to . However, now look what happens, well, the top I see 5 and in the bottom what do I see? I see the x which is just 3. So now I see something that equals the original thing of , but notice now that those discriminating people would be reasonably happy because the denominator now no longer has a square root in it.
This is often called clearing a denominator for a radical. And that method is if it is something very simple like this is you just multiply top and bottom by one and the choice of one is pretty straightforward it’s just the square root. For example, let’s try another one just to really drive this home. If I have and I want to clear the denominator. What would I do? I would multiply top and bottom by the same thing, so I’m not going to change anything, the key of course is you don’t want to change this number, and what does this equal. Well, this equals 2 . And what do I see here? I see the x = 2. Hey, now look at that, these can cancel and I see the . So that is sort of interesting the is the same thing as . And now you can sort of see, gee, if I had a choice, I have to admit this is a lot nicer and more tidy rather than this which is sort of , who knows. So, sometimes in fact, making the denominator a rational, rationalizing the denominator is not necessarily a bad idea as you can see here.
Okay, these are easy examples. What about if we have something really sort of painful, for example what if you showed someone who was not very open minded that thing and you wanted to actually clean that denominator up and have no square root down there. Well, what would you do? Well, you see a great first guess would be to multiply the top and the bottom by that thing, it worked before. This is a fantastic guess, you know, making guesses is the most important thing you can possibly do in mathematics, whether they are right or wrong. If we try this we are going to see something very unfortunate. When we actually multiply all this bottom stuff out we are going to see we are going to have more square roots there. Let me show you this really, really quickly and I’m going to go through this in a little slower detail in a moment.
If I were to multiply the bottom out, you see here I would still have 3 . But that bottom, well, I have to actually either use the foil method of foiling everything out, the first times the first, the outside terms, the inside terms, and the last terms or just distribute everything over. But, the x = 2, the inside terms give me a + , the outside terms give me another + , and the last terms give me a 1. So when I combined all that mess, I see 2 + 1 = 3. But unfortunately I see + and that is 2 . I lost ground, because now I have square roots on top and bottom, not good. So, how do you get around this? Well, you get around this by actually a very clever trick, which turns out to have a very pretty solution.
And that trick is when you have something complicated like this, not just a square root alone, but a square root in some other stuff, the trick is to multiply top and bottom not by that exact thing, but by the exact thing where you switch the sign. Whatever sign this is, make this the opposite. In this case ’m going to actually multiply top and bottom by . And this object, first of all, it is called the conjugate, see conjugate. And the conjugate is just what I said, if you have a sum square root of something and the conjugate is the exact same thing with the sign between them reversed. Let’s see what happens when we multiply now by the conjugate?
So same question, I’ve got . But now I am going to multiply by a different choice of 1. I am going to multiply by the conjugate. Watch what happens. So top and bottom by the -1, see the conjugate has the opposite sign. Okay let’s go. So here I see three times this, divided by and now I’ve got this thing here +1 times -1. Are you happy with what I have here? Well, I hope not, because actually there is something that is not right. What’s not right? You see, if you look at this, just forget everything else in the world, look at this and what do you see? You see 3 -1. Is that what we really want to say? Absolutely not.
What we want to say is 3 times this whole thing. You have to always remember that if you are multiplying 3 by a whole bunch of stuff you’ve got to keep those parentheses right in there. This is really a classic mistake and the idea is that you have to always distribute, always distribute. So now let’s keep going. If I distribute that, that would be 3 -3. And now let’s actually work out this. Now that is going to require a foiling thing, so remember foil, there are logical ways of thinking about this. Foil is just a little short hand for thinking first times first, the outer terms multiply them, the inner terms multiply them, the last terms multiply them and then add everything together. If you don’t like that you could just think about it as what it really is, which is distributing. Just to take this whole thing and multiply it by this and then multiply it by that and then distribute again this way. So in fact you get a whole bunch of terms that multiply. The x we know is 2, now the inside terms give me 1 x so that is just + , but look what happened to the outside terms I get a -1 x , which is - and now you can see the magic of what is happening here. Notice I have a , but then a - they actually annihilate themselves this is actually looking promising. And then a 1 x -1 is just a -1. So look what happens, these guys kill each other and I see 3 - 3 all divided by and then I have 2 - 1, and 2 - 1 is just 1, so I see an answer of 3 - 3. Or if you’d rather if you factor it out, either answer is of course great. But, you can see the power of this just by taking this thing, which looks pretty ugly, I can see it is actually equal to something much nicer. So this thing of rationalizing denominators is not always a crazy idea.
Let me try one last thing because I want to show you, this one I’m just going to do for you, I want you to sit back and enjoy it. But it is one of these questions where at the end of the problem I want to know if you are happy or if you are sad. So here we go. Here’s the problem; I want to multiply this thing to clear the denominator. So what do I do? Well, I multiply top and bottom by the conjugate of the denominator. So what I have to do here is multiply the top and bottom by, well, what’s the conjugate? Okay, here we go it is going to be this thing where I changed the sign, . And I have to do the exact same thing on the bottom otherwise that equal sign will be false, because the point is for this thing to be equal I’m just multiplying by one, which does nothing. Okay, multiply out the bottom what do I see? I see a 1 and then the inside term and the outside term cancel and I’m left with - x , which is just -3. And what do I get on the top? Well, the same thing, I get a 1, the terms cancel and the last term is + 3, so this looks like , which equals -2. There you go.
Now the question is are you happy? Well, I hope you’re really not happy, because actually I made a big mistake. And the big mistake is actually right here that actually is false. The bottom was fine, but with the top I’m actually going to have square roots there. The truth is there was no cancellation. If you enlarge this to show detail what we would really see is the following, let me just pick up this little part. If we do it really carefully 1 x 1 = 1, and then I have , but see then I have another and then finally I have a + 3. So in reality what I see is a 1 + 3 = 4 and a + = 2 . And so the reality is I have 4 + 2 on top, that’s the top. And so the actual answer would be what? If I pick up the action here that would equal the top, 4 + 2 all divided by what we got before which was the correct bottom, -2. And so we could simplify this a little bit if I factor out the common factor of 2 here and cancel. I think I’d see the following, I would see , which I could write as just . Well, I’m done with this problem and as you can see, my ink is running out so I better stop talking.

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