Int Algebra: Rewriting Complex Fractions

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Rationals
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Now, dealing with complex rational expressions is the exact same thing as dealing with regular fractional complex expressions, namely, we have to sort of clean things up and make the fractions all fit together. I want to jump right to an example, so you can really see that this is nothing incredibly new, but there are some tricks of the trade here that you might sort of want to think about.
So this is a complex rational expression, because it’s fractions on top of fractions, there are variables, there’s just a big old mess: t2 2t2 2+?. You might want to think about inverting and multiplying, but you can’t do that here, because you only invert the whole bottom. So you have to get a common factor here and invert and multiply.
There are two ways of dealing with this kind of thing, and let me talk to you about both of them really fast, and then I’ll illustrate some of them. One way is to do exactly what I just advertised, namely, get a common denominator and make this as one fraction, one fraction and then invert and multiply. Let’s think about that for a second, and I’m going to do it for you really fast, just so you can get the sense of this. If you were to work that out, getting a common denominator here, I just multiply top and bottom by t. So this would be t2t. So I’d see t2 - 2t. That’s just the top there with the common denominator. Now, you see how that’s just a 2? If I canceled, that’s just two. It’s all divided by – I’ll do the same thing here. I would see t2 2t+. And now you can happily invert and multiply. And if you do that, what would you see? You would see the following: you would see 22ttt2 - 2t+?. Now you can notice the common factor of t that could cancel out. And, in fact, you can see I can pull out a factor of 2 here, which would give me 2t – 1, and a factor of 2 here, which would give me 2t + 1, and then the 2’s actually can cancel out. So it’s sort of messy, but the bottom line is this thing equals, very simply, 1 t1 - t+. So this is one way of doing it, namely, just getting a common denominator here, combining this, making it one big fraction, making this one big fraction, and then inverting and multiplying. That will work.
Another way to do the same problem, the same type of thing, is to look at this and try to clear the denominators off the top and bottom by a very clever choice of multiplying top and bottom by the same, namely, by multiplying by 1. And notice that if I multiply this whole thing by t, that would clear the denominator. And the same t works here. So, in fact, if I just multiplied top and bottom here by t, so that’s tt, which doesn’t change the value of anything, then what do I see? I’ve got to distribute. Don’t make a classic mistake. I’ve got to distribute that t everywhere. And when I distribute that t here, I see 2t, and when I distribute that t here, I see a cancellation, so I'm left with –2. On the bottom when I distribute, I see a 2t, and then here I see a plus 2. You see the common factor of 2, as I saw before, 1 2t1 - 2t+. A little cancellation and I’m left with the answer we got before, 1 t1 - t+. I think actually a little bit easier, because I didn’t have to get common things and do things and combine and invert and multiply. I just had to find a multiple that would cancel out all the bottoms. So that’s another way of doing it.
Rationals
Operations with Rationals
Rewriting Complex Fractions Page [2 of 3]
Let’s take a look at another example to illustrate the point. Suppose we have 2x1 x11?+1. Again, two methods to do this. I could get a common denominator here, multiplying the 1 top and bottom by xx and multiplying this 1 top and bottom by 22xx, getting one big fraction here and one big fraction here and multiply. That will work. But let’s think about this least common multiple thing.
What’s the smallest thing required for me to kill off all the denominators? Well, all I need here is an x. But here I actually need an x2, so I’m going to multiply top and bottom by x2. If I do that, 22xx doesn’t change the value of anything. That’s just 1. So what do I get? When I distribute, I see x2 plus – and if I distribute xx2, it’s just x. Now on the bottom, what am I left with? I’m left with just an x2 minus – and then here I just have a 1, because they cancel.
So that’s certainly a fine answer or, if you really got into it, you realized I could factor all over the place here. Let’s factor out the common factor of x on top, which will leave me with an x + 1. And notice that the bottom is a difference of two perfect squares. And look! We can actually cancel away the x+ 1. And so if someone said, “I want this in lowest terms,” this probably wouldn’t cut it. I’d have to say 1 -x x, and that’s in lowest terms.
So this complicated-looking thing that we first saw turns out to be just little old 1 -x x. Who would have guessed it? Anyway, let’s try one last one here. In fact, I’ll give you the opportunity to try this one. x1x11 x 1?+. All right, give that a try and see how you make out.
All right, well there’s a couple of ways of doing this. You know, actually, this is just one fraction here, so the method of inverting this and multiplying might not be a bad idea here, because if I just invert that and multiply, it might simplify things nicely. The other possibility again is to get the least common multiple. In this case, the least common multiple would be the following: let me just tell you what it would be, in case you did it this way. It would be x(x + 1). So I would multiply the top and the bottom by this. And if I did that, I’d cancel all the denominators. See, I’ve got to cancel that thing and that thing. This just wouldn’t cut it, because that would not be able to cancel with the x. I need to have a factor of x there. So this is the least common multiple of all these people.
But actually, just for fun, I’m going to show you the other way of doing it. Notice this is just one big fraction here, so let me invert it and multiply it and see what happens. So if I invert and multiply, now it’s 1x. So I'm actually just multiplying this whole top by x. So this just becomes x times all that. So what would x times all that be? Well, if I distribute the x, what I would see is an x on the top there minus – and if I distribute the x here, what do I have? xx; I have an x here multiplied, put an x on the bottom, I’d just get 1. So in fact, it equals that, which would be a great
Rationals
Operations with Rationals
Rewriting Complex Fractions Page [3 of 3]
answer. Of course, if you want to get a common denominator, now it’s easy to do it. I just multiply top and bottom here by x + 1 and I would see 1x1x1 x x++?+ . I just turn that 1 into 1 x 1 x ++ and subtract. If I subtract, what would it look like? I see an x – x = 0. And then it looks like I have just that 1 there all over the common bottom. So 1 x 1+. Is that what you got? I hope not, because that’s wrong. Classic mistake number four, once again, that’s right, it’s the subtraction mistake. Don’t forget that you’ve got to share the negativity. When I subtract, I’ve got to subtract everybody, and that negative sign has to hit here, as well as here. So, in fact, this is missing the necessary negative sign to make this correct. So it’s 1 x 1-+. That’s the answer.
Okay, so complex rational expressions, a little bit of a deal, I’ll grant you that, but not a big deal. You’ve got to be really careful. Either find a least common multiple to clear out those denominators, or just combine everything, have one big fraction over one big fraction, invert, multiply. Be really careful with the little bit of arithmetic there and you're home free.