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Simplifying Complex Fractions - Part 3

Looking at this fraction is a little bit different than looking at the other complex fractions that we've dealt with. However, we can make it look like the other complex fractions. But first of all, we have to know what these negatives actually do to this. Okay? So before we move on, let's think about that and go through an explanation. Let's go over here to a nice, clean sheet. We have x to the negative 1 equals 1 over x. Basically, this means that anytime that you have a negative exponent in the numerator, you can easily bring that down to the denominator to make it have a positive exponent. That's what happens with all of these. If you have a negative exponent in the numerator and you bring it down to the denominator, you have a positive exponent. If you were to have a negative exponent in the denominator, and move it to the numerator, then that exponent becomes positive. So basically what happens is when you switch it from numerator to denominator or from denominator to numerator then it changes the sign of only the exponent and nothing else. Even if you had something like 2 to the negative 1. We know that there is a 1 underneath this 2 - that's implied. So if we were to move this down to the denominator, then we’d have 1 divided by 2 to the first. Which is really, of course, just one-half. Likewise, if you had something like 1 over 3x. And let’s say that  that’s actually in negative 3 – let’s raise it to the power of negative 3, so that when you move it up, remember, it's the whole entire quantity, so when you move it up, you actually have 3x in that quantity raised to the third (and, of course, it's over 1). So remember it can simply be written as 3x to the third. Or if you want to go even further with it, you could distribute this 3 through, so you could say that it’s going to be 27x cubed, if you’d like. That's going to be another video, so if you want to, please scroll videos and search for the ones you need to know especially about this idea here. So, with that, let's go ahead and again look at this problem. Now that we have these negative exponents, we know exactly what they mean and we can change them into that. So, let's do. We have 1 over x divided by 1 over x squared plus 1 over y squared. See how I just moved everything from the numerator to the denominator thus changing the signs of only the powers. Okay, so now, this is much like the previous video where we had one fraction up here in the numerator and notice now we have two fractions down here. So we need to make this into one fraction. And we simply do that by finding the least common denominator, which I’m pretty sure you’re familiar with by now. So, we’re going to have that LCD of x squared times y squared. And so, we need to multiply this guy by y squared divided by y squared so we can get a y squared here. And we need to multiply this guy by x squared over x squared, so we have plus x squared. So now notice, we have a fraction here at the top, and a whole fraction here in the denominator. So now all we have to do is keep the idea of keep, change, flip. So, we’re going to keep the first fraction, which is 1 over x. We’re going to change the sign from division to multiplication. And we’re going to flip the last fraction, which would be x squared times y squared, divided by y squared plus x squared. Notice, there are two x’s in the top, and one x in the denominator, so what we’re going to do is we’re going to cancel that out, and that’s actually going to be just x to the first. So we now have, just multiplying straight across the top, we’ve got x y squared, all divided by y squared plus x squared. And if you’d like, you can make this bottom part have the x squared come first. It doesn’t matter, because you have to remember that “a” plus “b” equals “b” plus “a.”

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