### Rational Exponents

- When an exponent is a fraction, it can be re-written as a root and a power. The numerator of the fraction represents the power, where as the denominator represents the index of the root.
- All of the rules for exponents apply when the exponent is a fraction. This means we can use many familiar rules to work with radicals.
- Remember that when a variable is raised to a negative exponent, it can be written as 1 divided by that quantity.

### Rational Exponents

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Objectives 0:07
- Rational Exponents 0:32
- Power on Top, Root on Bottom
- Example 1 1:37
- Rational Exponents Cont. 4:04
- Using Rules from Exponents for Radicals as Exponents
- Combining Terms Under a Single Root
- Example 2 5:21
- Example 3 7:39
- Example 4 11:23
- Example 5 13:14

### Algebra 1 Online Course

### Transcription: Rational Exponents

*Welcome back to www.educator.com.*0000

*In this lesson we are going to take a look at rational exponents. *0002

*The neat part about rational exponents is you will see that we will develop a way and connect them to our radicals.*0011

*You will see that there are a few rules for working with these rational exponents and a lot of them come for just our rules for exponents.*0019

*We will look at a few ways that you can combine terms that have some these rational exponents on them.*0027

*We have seen many different types of radicals. *0034

*We have seen exponents but there is actually a great connection between the two. *0039

*If you have a radical of some index like a square root or third root and it is raised to a power, *0044

*you can write this in one of the following two ways.*0051

*You can write it as the root of that expression ratio power or you can write it as the expression raised to a fractional power. *0054

*One thing to notice were the location of everything has gone to.*0066

*The power in each of these problems I have marked that off as a that shows up on the top of the fraction. *0070

*The root is going to be the bottom of the fraction.*0083

*You can take any type of radical and end up rewriting it as a fractional exponent or as a rational exponent.*0089

*To get some quick practice with this let us try some examples.*0098

*I have 36^Â½, -27 ^{1}/3, 1^{3}/2 and -9^{3}/2.*0102

*We are going to evaluate these up by turning them into a radicals. *0110

*Okay, so this first one, the way we interpret that is that I'm going to put it under a root with an index of two.*0116

*This is like looking at the âˆš 36, which is of course just 6.*0123

*For the other one, if I see -27 ^{1}/3 that will be like taking -27 to the 3rd root so this one is a -3.*0132

*In the next one notice how we have a power and a root to deal with. *0146

*81 ^{3}/2*0149

*There are two ways you could look at this.*0153

*You could say this is 81 ^{3} and we are going to take the square root or you can take the square root of 81, *0155

*then raise the result to the third power.*0163

*Both of them would be correct, but I suggest going with the one that is a lot easier to evaluate.*0166

*Iâ€™m thinking of this one. *0171

*If you take 81 and raise it to the third power we are going to get something very large *0173

*and try to figure out what is the square of that is going to be a little difficult.*0177

*But look at the one on the right, I can figure out what the âˆš 81 is, I get 9.*0180

*We can go ahead and take 9 ^{3} it would be 729.*0189

*One last one, I have 9 ^{3}/2 and a negative sign out front.*0201

*Let us first write that as the âˆš9 ^{3} as for this negative sign it is still going to be out front.*0206

*I have not touched it.*0215

*I'm not including that in everything because there is no parenthesis around the -9 in the original problem.*0217

*Iâ€™m starting to simplify this.*0224

*The âˆš9 would be 3 then I will take 3 ^{3} and get -27.*0226

*In all of these situations I'm looking at the top of that fraction make it a power, we get the bottom to see what the root needs to be. *0236

*The good part about taking all of our radicals and writing them using these exponents, *0247

*it means that we can use a lot of our tools that we have already developed for exponents and we have done quite a bit of them.*0252

*In fact a quick review, we have a product rule for exponents, a power rule *0258

*and we have gotten different ways that we could go ahead and combine them. *0264

*We also have rules on how to deal with fractions like adding subtractions, subtracting fractions. *0269

*We have our zero exponent rule, our quotient rule and negative exponent rules. *0273

*All of these rules will help us when working with our radicals.*0278

*Just watch on what our base is and what we need to do from there. *0284

*Now using some of these rules if you do have radicals you might be able to combine them under a single root.*0293

*That involves using a common denominator most of the time.*0305

*You can use this tool for working with rational expressions.*0311

*Watch how I find the common denominator for some of my problems and actually get everything under one root.*0315

*Let us do these guys a try.*0323

*64/27 ^{-4}/3 *0325

*Before I get to that 4/3 part Iâ€™m going to apply some of my other rules for exponents and specifically that negative exponent. *0329

*One way I can treat a negative exponent is it will change the location of the things that it is attached to.*0339

*Iâ€™m going to write this as (27/64) ^{4}/3.*0344

*Iâ€™m going to give that 4/3 to the top and to the bottom that is using my quotient rule.*0352

*I will end up rewriting the top and bottom using my radicals. *0362

*Iâ€™m looking at the 3rd root of 27 and we will take the 4th power of that *0369

*and then we will take the third root of 64 and take the 4th power of that one.*0376

*Let us see what this gives us.*0390

*On top the 3rd root of 27 that will be 3 and 3rd of 64 =4 and both of these are still raised to the 4th power.*0393

*We will go ahead and take care of that multiplication.*0403

*81 / 256*0407

*Being able to take it and write it as radicals, it meshes well with all the rest of our rules.*0413

*Let us use the quotient rule on this next one.*0421

*4 ^{7}/4 Ã· 4^{5}/4*0423

*I need to subtract my exponents.*0429

*Good thing both of these have the same denominator this will simply be 4 ^{2}/4.*0434

*That continues to simplify this would be 4^Â½ which written as a radical is the âˆš4 which all simplifies down to 2.*0442

*Be very comfortable with switching back and forth between those rational exponents and your radicals.*0451

*On to ones that are a little bit more difficult. *0461

*These will involve trying to simplify much larger expression and some of the terms will have those rational exponents. *0464

*Okay, so here I have (r^Â¼ y ^{5}/7)^{28} Ã· r^{5}.*0474

*I can apply my 28 to both of the parts on the insides since they are being multiplied.*0486

*Let us see what this looks like.*0493

*28 Ã— 1/4 = 28/4 and I have y ^{5} Ã— 28 Ã· 7 and all of that is being divided by r^{5}.*0495

*Let us see if we can simplify some of those fractions.*0512

*How may times this 4 go into 28? 7 times and 7 will go into 28 four times.*0515

*This is (r ^{7} y^{4})/r^{5}*0527

*We can go ahead and reduce our y.*0534

*5 of them on the bottom and with 5 of them on top that will leave us with an r ^{2} and y^{4}.*0536

*Let us try another one.*0548

*This one has a lot of fractions and a lot of negative signs.*0549

*(P ^{-1}/5 q^{-5}/2)/(4^{-1} p^{-2} q^{-1}/5)^{-2}*0553

*Iâ€™m going to use my rule to apply this -2 to all of my exponents.*0565

*Iâ€™m sure that will help get rid of a bunch of different negatives.*0572

*-2 Ã— -1/5 = 2/5, -2 Ã— -5/2 = q ^{5}*0576

*Then on to the bottom, -1 Ã— -2 = 2 and p ^{4} = 4, q^{2}/5.*0593

*That does simplify it quite a bit.*0610

*At least I can see that this guy right here will be 16 but we will also have to reduce these a little bit more. *0611

*I'm going to take care of these ones I want to think what is 2/5 â€“ 4.*0620

*If we can find a common denominator it will helps out with that ones.*0626

*2/5 - 20/5 we will call that one -18/5, so I know that I will have 18/5 on the bottom.*0629

*Let us try it out.*0646

*I have a 16 on the bottom and now I discovered I have a p2 ^{18}/5 on the bottom as well.*0648

*These ones we can reduce. *0656

*I want think of 5-2/5.*0659

*The common denominator there will be 5, 25/5 - 2/5 = 23/5 we will put that on top q ^{23}/5.*0663

*We have our final simplified expression. *0679

*In this next two I have some radicals and we will go ahead and write them as our rational and see what we can do from there.*0686

*The top, this would be y ^{2}/3 Ã· y^{2}/5*0696

*If Iâ€™m going to end up simplifying using our quotient rule, we will look at this as 2/3 â€“ 2/5.*0705

*We need a common denominator on those fractions to put them together.*0713

*Let us look at this as over 15.*0721

*10/15 â€“ 6/15*0726

*This would give us y ^{4}/15.*0731

*Then we can continue writing this as a radical if I want it to be the 15th root y ^{4}.*0735

*That is what I was talking about earlier about being able to combine these radicals into a single radical.*0744

*Let us try this other one.*0752

*I have z and Iâ€™m looking for the 5th root of it, I will write that as z ^{1}/5.*0753

*When Iâ€™m taking the 3rd root of all of that, that is like to the 1/3.*0761

*My rule for combining exponents in this way says I need to multiply the two together.*0769

*Z ^{1}/15*0774

*If I will write this one as a radical them Iâ€™m looking at the 15th root of z.*0779

*There are many of the different examples that we can get more familiar with using these radicals and rational exponents.*0786

*Let us do one where we work on writing it using these radicals.*0796

*Iâ€™m looking at the 8th root of the entire 6z ^{5} â€“ 7th root 5m^{4}*0805

*Be careful when it comes to trying to combine things any further from there.*0824

*We have not covered yet on what to do with addition.*0828

*Most of our rules cover our rules for exponents you got to be careful on what you do with addition.*0834

*Iâ€™m going to leave this one just as it is and not work on combining.*0843

*One problem that I will have is that my bases are not the same in anyway.*0849

*This is good as it is.*0855

*Be very more familiar with taking the radicals and turning them into rational exponents.*0858

*Remember that the top will represent the power and the bottom of those fractions will represent the index of the radical.*0864

*Thank you for watching www.educator.com.*0871

1 answer

Last reply by: Professor Eric Smith

Mon Aug 26, 2013 2:57 PM

Post by Norman Cervantes on August 24, 2013

8:48 i think you forgot to multiply 5 with 4.