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Lecture Comments (2)

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.

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

Transcription: Rational Exponents

Welcome back to

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 36^½, -271/3, 13/2 and -93/2.0102

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 √ 36, which is of course just 6.0123

For the other one, if I see -271/3 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


There are two ways you could look at this.0153

You could say this is 813 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

I’m thinking of this one.0171

If you take 81 and raise it to the third power we are going to get something very large0173

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 √ 81 is, I get 9.0180

We can go ahead and take 93 it would be 729.0189

One last one, I have 93/2 and a negative sign out front.0201

Let us first write that as the √93 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

I’m starting to simplify this.0224

The √9 would be 3 then I will take 33 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 rule0258

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


Before I get to that 4/3 part I’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

I’m going to write this as (27/64)4/3.0344

I’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

I’m looking at the 3rd root of 27 and we will take the 4th power of that0369

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 / 2560407

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

47/4 ÷ 45/40423

I need to subtract my exponents.0429

Good thing both of these have the same denominator this will simply be 42/4.0434

That continues to simplify this would be 4^½ which written as a radical is the √4 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^¼ y5/7)28 ÷ r5.0474

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 × 1/4 = 28/4 and I have y5 × 28 ÷ 7 and all of that is being divided by r5.0495

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 (r7 y4)/r50527

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 r2 and y4.0536

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)-20553

I’m going to use my rule to apply this -2 to all of my exponents.0565

I’m sure that will help get rid of a bunch of different negatives.0572

-2 × -1/5 = 2/5, -2 × -5/2 = q50576

Then on to the bottom, -1 × -2 = 2 and p4 = 4, q2/5.0593

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 – 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 p218/5 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 q23/5.0663

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 y2/3 ÷ y2/50696

If I’m going to end up simplifying using our quotient rule, we will look at this as 2/3 – 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 – 6/150726

This would give us y4/15.0731

Then we can continue writing this as a radical if I want it to be the 15th root y4.0735

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 I’m looking for the 5th root of it, I will write that as z1/5.0753

When I’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


If I will write this one as a radical them I’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

I’m looking at the 8th root of the entire 6z5 – 7th root 5m40805

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

I’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