 Welcome, this is another video about exponents, but this one is now about rational exponents, so we have to deal with some fractions. Alright, the first problem we're going to be working on here today says r to the 1 seventh divided by r to the 2 thirds. So just like we did before, because we're dividing, we have to subtract our exponents. So that's going to look like r to the 1 seventh minus 2 thirds. And I wrote those large so you can see them, but they are both in the exponent. Okay, now, because we're subtracting two fractions, we need to find a common denominator. So that would look like r to the, while our common denominator is going to be 21, if you multiply 7 by 3, that gives you 21. So multiply the 1 over 7 by 3 over 3, so that's going to give us 3 over 21 minus, multiply the 2 over 3 by 7 over 7 to build up to 21. So this would be 14 over 21. Okay, so now that we have our common denominator, we can subtract. So that gives us a grand total of r to the negative 11 over 21. But again, when I read the directions, it's that all exponents should be positive. So that means our entire piece here then needs to get moved to the denominator. Some of you may be tempted to do the reciprocal of the exponent, but that doesn't make your whole fraction then, or the whole expression flip. So you want to do 1 over r to the 11 over 21. And that would be a perfectly legitimate answer. Or you can also write this if you really want to be crazy. You could write this as 1 over the 21st root of r to the 11th power. Okay, so either way, it just depends on how your instructor wants you to write it. But I am perfectly happy with your answer being left like that. Okay, problem two. This one has some coefficients as well to deal with. So we've got 8x to the third, y to the ninth, all to the 1 third power. So just like we did before, we need to distribute our power to all three of these pieces. Okay, so I'm going to write this out in detail. So we have 8 to the 1 third times x cubed to the 1 third times y to the 9th. Let's see, to the 1 third. That's another problem, sorry, to the 1 third, there we go. Okay, so 8 to the 1 third, that means you need to do the cube root of 8. Or you can do that on your calculator as well. Or ask yourself, what times itself three times will give me 8? And then answer is 2. Like we did before, powers to powers, that means we multiply. So three times a third is going to be, well, three times one is three divided by three gives us one. So that'll give us an x to the first, y to the ninth to the 1 third. Again, you have to do a power to power, you multiply. So nine times a third gives us nine times one, which is nine divided by three, which is three. So this would be y to the third. So this answers good, or if you'd rather, you can just write it as 2xy to the third, since that x has a one for the x one. Okay, fantastic. The next problem is very similar to this one, but it's got two of these pieces to it. So I'm going to do this a little bit more quickly and not written out quite as much as I did here. So again, you have to distribute your power in your numerator. So you're going to have to distribute the half power to the 81, to the r to the six, and to the 20th power, s to the 20th power. So that's going to look like, so 81 to the one half power is nine. Because the square root of 81 is nine. Or you can ask yourself what times itself two times will give me that 81, and that's a nine. Okay, then we're going to have r to the six to the one one half. So again, because we have a power to a power, you want to multiply. So that's going to give us r to the third. And s to the 20th to the one half power, power to power, you multiply. So that'll give you s to the 10th. So this is where I want to make a note. You have to remind yourself you do different things with coefficients that you do with powers. So coefficient, you do the power of that. But exponent, you end up multiplying. Okay, now in the denominator, again, we need to do the same thing. This one looks a little bit nicer because it's not a fraction. So I'm going to do s or three to the second power. So that gives me a nine as well. R to the fourth to the second power. So power to power we multiply, that's r to the eighth. S squared, gives me s to the sixth. Okay, now we have this fraction. We just keep simplifying because I see two r's and I see two s's. I also see two nines, so let's go ahead and wipe those out. If you want to put a one out front here, you can for your coefficient or just ignore it. Then now let's take a look at these r's. So I see an r to the eighth and I see an r to the third. My r to the eighth is in the denominator, so that one's obviously bigger. So I'm going to subtract these and make my fraction. If I subtract three minus eight, that gives me negative five. So that means my r to the fifth is going to have to go in the denominator. Okay, again, you can also look at this as the eight is in the denominator. That one's bigger, so that's how I know that's where my r goes. Now, for the s's, I've got an s to the tenth and I've got an s to the sixth. My s to the tenth is in my numerator. That one's a lot bigger again, so that means my s is going to go in my numerator. And when I do ten minus six, that gives me a grand total of four. Okay, so that's my final answer. I'm going to subtract the fourth over r to the fifth. And last, but certainly not least, we have a big ugly fraction to a big ugly fraction power. Well, this one sounds so big, but you know what I mean. So what I would probably do again, like we've done before, is I would simplify what's in here first, so that way it makes stuff smaller. And then we can go ahead and do our power. So I see a nine. Okay, I don't think that one can get simplified anymore. Let's see, so we have a nine here, and that's going to go in the numerator. Make my division bar. We have an x to the fifth and an x to the third. Okay, so I'm going to highlight those, so I've got my x's here. So I want to subtract. My five is bigger, so that one's going to go in the numerator. So that's going to give me an x squared in the numerator. I have y to the fourth, and I have y to the negative two. So the four is bigger, this is going to go in the numerator. So I guess I really didn't need my fraction bar, but that's okay. We didn't know that ahead of time. And then I can subtract these. So four minus the negative two is going to give me y to the sixth. Another way you could have thought about these y's is this one here has the y to the second has a negative exponent. So that's going to technically get moved up to the numerator. So I'd have y to the fourth, y to the second, which would give me a total of y to the sixth. So either way you look at it, that's the answer you get. Okay, this is over one, I guess, if you want to make a fraction out of it. You don't have to though. All right, so one half power, again, I have to distribute that through. So distribute here, here, here, and the one, you know, one anything's just one. So nine to the one half power is three. We did that one early. Oh no, we didn't do that one earlier. So three squared gives me nine, or the square root of nine is three. So whichever way you want to look at it there, x squared to the one half power. So two times a half gives me a one in six, or y to the sixth to the one half. Six times a half gives me a three, so that's going to be y to the third. Again, because this denominator was one, I don't even have to put that on here. And then because this is the next to the one, I think I'm going to write it a little bit neater as three x, y to the third. Okay, so hopefully you guys can see by looking at these problems, the fractional exponents is just like what we did before with the whole number exponents. So it's nothing different, it's just a little bit uglier. And you have to think roots instead of stuff, which means stuff gets smaller instead of powers, which means stuff gets bigger. Okay, keep practicing on these, and good luck with your problems.