Simplifying complex fractions with negative exponents; So the trick for solving this kind of problem is to first rewrite this without negative exponents and for that I want to talk a little bit about a reminder on what all these negative exponents mean.
Okay so quick review, x to -1 is the same thing as 1 over x. Y to the -2 is the same thing as 1 over y². So when we add these two things together, if I say x to the -1 plus y to the -2 this is just going to be 1 over x which is our x to the -1 plus 1 over y².
A common mistake that kids like to do is to combine these two into one fraction right away and what I mean by that is go from this to 1 over x plus y². They just say you see these negative exponents they say okay goes to the denominator and we’re done with it. But if you think about it, how do we combine these two fractions. In order to combine fractions we need a common denominator. So if we were actually to multiply this out it would simplify this up we would have something in the numerator, so this actually is not right at all. We first need to write them out individually and then if we want to combine them we would have to get a common denominator. So that said let’s take a look back at this problem.
What I have here is 3x to -1. Let me rewrite that over here just a little emphasis, this is 3 times x to the -1, x to the -1 as we just talked about over here is 1/x, so this is actually 3 times 1/x. The -1 is not associated with the 3 so the 3 doesn’t go to the bottom. What this term actually is then is just 3/x. If you had a parenthesis around the 3 and the x then the 3 would go down to the bottom as well. But it’s not there so the 3 is going to stay at the top. Minus 3y to the -1. Same exact thing we did right here except we replace the x with the y, 3/y. Over y to the -2 we just talked about it over there, 1 over y² and then minus x to the -2, 1 over x².
Now we have rewritten our problem, gotten rid of all our negative exponents and now have just as a complex fraction with positive exponents. From here there is two methods we can solve it. We could either combine our numerator and denominator separately or multiply by the LCD of the entire thing. Either one’s fine, in general I prefer to multiply by the denominator of the entire thing so that’s what I’m going to do but if you want to do it the other way that would work just as fine.
We need to look at this and see what the least common denominator is of the entire thing. It’s going to be, we have two xs and two ys, xy it’s going to be x²y². So we multiply the entire thing by x²y²,and I can just multiply everything by the same thing because that’s the same thing as multiplying by 1 which doesn’t change our problem. Okay, so let’s multiply this out. Let’s get rid of our little refresher course over here and distribute these x²y² to the problem.
For the first term one of our xs is gong to cancel leaving us with 3xy². For our second term one of our ys is going to cancel leaving us with 3x²y and in the denominator one of our first terms are ys, our y² is going to cancel leaving us with x² and lastly minus y². Factor and simplify.
So looking at my numerator everything has a 3xy leaving me with y minus x, denominator is difference of squares so this is going to be x minus y and x plus y. Now what we have are two things that look very, very similar. We have an x minus y and a y minus x. These are opposites, okay? Whenever we have opposites, same numbers just opposite signs, these cancel to be -1. So these cancel to be -1 leaving us with -3xy all over x plus y.
To recap; when solving a problem when we’re dealing with complex functions and negative exponents always rewrite your exponents as positive and then either combine your terms and combine your numerator and denominator, flip and multiply, or multiply through by the LCD of the entire thing and simplify.