Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Complex rational expressions. When I have a fraction divided by another fraction, what that really means is it's the first fraction divided by the 2nd fraction. Or we could think of it as A / B * D / C or AD over BC. Sometimes the means and the extremes is the reference, the extremes being the AD and the inside being the BC. So what we're going to do is we're going to treat this top portion as a fraction until we get it to be a single monomial fraction, and then the bottom the same way. And there's a lot of different ways to solve this, but this is one method that will always work. So 1 / t plus the six is understood over one. So then we're going to make the common denominator be T. So it turns into six t / t, or finally just one plus 6T all over T So now what we're going to do is we're going to simplify that denominator portion. So this 1 / t -, 5 one over t -, 5 / 1. Actually, let's make that 5 / 1. So we get five t / t here. So 1 -, 5 T over T. So to actually finish this up, we're going to leave the numerator portion the same 1 + 6 T over T, and then we're going to flip the denominator and multiply. So t / 1 -, 5 T. Now the TS in the numerator and denominator that are by themselves are going to cancel. Remember, we could think about pulling out A1 if we wanted to make them into monomials. So then we can see that the 1 + 6 T and the 1 - 5 T are not exactly the same, so they can't cancel. So we get 1 + 6 T over 1 - 5 T as a final solution there. This next one, our common denominator for the numerator portion is going to be X - 2 * X - 1. So the first term is going to need that one times an X - 1, and the second one's going to need a three times the X -, 2. If we go ahead and look at the denominator, we're going to have the two needing to be multiplied by X - 2 and the 5 * X - 1 in order to get that common denominator of X -, 1 X -2. So now if we distribute, we get X -, 1 + 3 X -6 all over X - 2 X -1. I'm going to go ahead and flip it right now while I distribute, doing 2 steps at once to save a little bit of room. X - 1 * X - 2 and I distribute, I get two X - 4 + 5 X -5 S Combining, we get four X - 7. There's nothing in common, so we could think about pulling out of one if we wanted to. X - 2 X -1 X -1 X -2 When we combine down here, we get 2X plus 5X or seven X - 9 and they have nothing in common, so I could pull out of one there. Also, the X -, 1 X minus twos in the numerator and denominator will then cancel, leaving us a four X - 7 / a seven X - 9 for our final solution. A couple for you to try. And here when we have a negative in the exponent, it really means one over. So we have 1 / X + 1 / y / X ^2 -, y ^2, which we know is the difference of two squares. So we can go ahead and factor that into X minus yx plus Y over XY. So that numerator portion, the common denominator is going to be XY. So this one gets multiplied by X and the second one gets multiplied by oh, sorry, this first one gets multiplied by Y and the second one gets multiplied by X / X minus yx plus Y all over XY. So y + X over XY times, flipping it X / y * X minus yx plus Y. Now this y + X, we could think about pulling out a one to make it a monomial. So X + y and y + X are the same. And then the X * y and the X * y. So we end up with just one over X -. Y is a final solution. Number six is going to get a little longer. We're going to start by factoring everything. I think I'll come way over here if I can. So 1 / X - 1 X plus one plus 5 / X - 1 X -4 all over 1 / X - 1 X plus one and 2 / X + 1 X +2. So now common denominator in the numerator portion. Maybe I'll start way down here again. We're going to have this one times the X - 4 and the five times the X + 1 because I'm needing to look what they don't already have in the denominators. So X - 1, X plus one, and X - 4 is the common denominator there. If we look at the denominator portion, we're going to have the 1st 1 * X + 2 and then the numerator of the 2 * X -, 1. That's going to be over X - 1, X plus one, and X + 2. Distributing and combining like terms, we're going to get 6 X -4 and +5 is going to be a +1 all over X -, 1 X plus One X -, 4. And then we're going to multiply that by we're going to flip it so X -, 1 X plus One X + 2 all over 3 X +2 and a negative 20. Those will cancel positive two -2 so now when we reduce this up, the X -, 1 X plus one on top and bottom will cancel. So we're going to get six X + 1 * X + 2 over three X * X - 4. You have a couple to practice on. Thank you and have a wonderful day.