## Discussion

## Study Guides

## Download Lecture Slides

## Table of Contents

## Transcription

## Related Books

### Simplify Rational Exponents

- To simplify radical expressions we often split up the root over factors. If we are working with a square root, then we split it up over perfect squares.
- Since radicals follow the same rules as exponents, we can use the quotient rule to split up radicals over division.
- The square root of a squared number is always nonnegative.

### Simplify 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
- Simplify Rational Exponents 0:25
- Product Rule for Radicals
- Product Rule to Simplify Square Roots
- Quotient Rule for Radicals
- Applications of Product and Quotient Rules
- Higher Roots
- Example 1 3:39
- Example 2 6:35
- Example 3 8:41
- Example 4 11:09

### Algebra 1 Online Course

### Transcription: Simplify Rational Exponents

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

*In this lesson we are going to look at simplifying rational exponents.*0003

*The things we will look at are our product rule, quotient rule and what to do when they involve things like radicals.*0009

*We will also look at a few radicals that involve variables and how can we get into simplifying much higher roots.*0018

*If you have two non negative real numbers call them A and B.*0028

*And what you can do with them when they are being multiplied underneath a radical is put them under the same radical.*0033

*If those are already underneath the same radical and you are also free to go and separate them back out.*0041

*They are each underneath their own radical.*0047

*This rule right here is good for either simplifying or combining things.*0050

*You can use it in both direction, combine or break apart.*0055

*One very important thing to note is that this rule only works for multiplication,*0060

*do not try and use any type of splitting thing if you have addition or subtraction.*0065

*Where the simplifying part comes into play is by taking a very large number, large expression and breaking it up into those individual parts.*0075

*We can consider a radical simplified when there is no perfect square under the radical anymore.*0085

*Look at these different radicals, find the perfect squares that are in there and split up the roots over those perfect squares.*0093

*In a similar rule, we have that we can either combine things under radical when we are dealing with division*0106

*or if they are both under division, we can split up over the top and over the bottom.*0112

*This works as long as the value on the bottom is not equal to 0.*0118

*Of course that is because we cannot divide by 0.*0122

*Feel free to use this rule in both directions either combine or separate them.*0127

*Note that this rule only works for division.*0131

*Again, do not try and use this for addition or subtraction.*0133

*If we happen to have a variable underneath the root and it is raised to the power,*0140

*we can assume that variable represents a non negative number.*0143

*That will make sure that we are dealing with real numbers and not imaginary numbers.*0149

*The product in the quotient rule that we have just covered apply when all the variables up here under a radical sign as well.*0155

*We are dealing with non negative real numbers with these various different rules.*0162

*You can even use these rules for some much higher roots and when you are looking to simplify those it depends on the index.*0171

*For example, if you are looking to simplify a cubed root, then what we are doing is*0183

*we are trying to take out all the cubes that are underneath that root.*0187

*Here is a quick example.*0190

*Maybe we are looking at the 3√a ^{3} without simplifying it to just an (a).*0192

*In cube roots this happens to be true whether (a) is positive or negative.*0198

*As long as you are dealing with square roots, the square root of the square number is always non negative.*0204

*The product and quotient will be applied to both our higher roots as well as just square roots.*0212

*Let us practice a few of these rules with some example questions.*0221

*If you want to simplify some numbers like the √4/49 feel free to split it up over the top and or the bottom.*0225

*And then you can take each of these separately.*0234

*This would be 2/7.*0237

*Here us one where they already have a root over the top and the bottom but I cannot simplify them individually.*0245

*I’m going to use that the quotient rule in the other direction and we will put them underneath 1 root.*0251

*From there I can figure out how many times 3 goes into 48.*0259

*I have that 3 goes in the 48 at least 10 times.*0269

*I have 18 left over, that would be 3 × 6.*0277

*By combining them underneath one root it allowed to go ahead and simplify the numbers a little bit better.*0291

*I just have to look at √16 is just 4.*0296

*Whether you are putting things underneath the same root or separating them out sometimes they cannot be simplified.*0302

*In this next one I will look at the root of the top and the root of the bottom.*0311

*With the bottom I can take the √36 is just 6 but the √ 5 has to stay as it is because it is not perfect square.*0316

*Let us look at one more.*0331

*You want to look at the √3/8 × √7/2*0335

*With this one I have two different radicals.*0347

*Do not let those fractions distract you we are looking to combine them underneath the same radical.*0351

*I will just combine them together using multiplication.*0359

*Once I do that I could go ahead and continue simplifying from here.*0365

*This will be √21/60 and on the bottom 16 is a square number so I can even go further.*0368

*√21 / √16 will be √21 / 4*0379

*All of these examples are designed to get you more familiar with either taking that root*0386

*and putting over both the parts or combining them into a single root.*0390

*Let us try a few that have some variables in them and notice how the same process works out.*0397

*What I want to do is simplify the √x ^{6} and I could imagine splitting up my x^{6} into lots of x^{2}.*0404

*If I use my rule to split up the root over each of those I will have √x ^{2}, √x^{2}, I will have that three different times.*0419

*Each of these would simplify to just an x.*0429

*I can go ahead and package that altogether as x ^{3}.*0433

*Notice how that fits with some of our other rules such as rewriting this as x ^{6} / 2 which is x^{3}.*0436

*All our rule stays nice and consistent with one another, they both agree.*0448

*Let us try another one.*0454

*We will slip this one up over 100 and over p ^{8}.*0457

*I could think of the √100 that is 10 and then for the p ^{8} that could be (p^{4})^{2}.*0465

*That square and square root can take care of each other 10p ^{4}.*0475

*That one will be good.*0481

*Let us try another one. This one is √7/y ^{4}*0485

*I will put the square root on the top and in the bottom.*0492

*We want to look at that y ^{4} as (y^{2})^{2}.*0498

*In that way we can see that this square and square root will take care of each other.*0505

*I’m left with the √7/y ^{2}*0511

*In this higher root the same rules apply.*0523

*When it gets down to breaking them up and simplifying them, nothing changes.*0526

*We just have to be worried about looking for our cubed numbers or in some of these other examples a number raised to the fourth power.*0531

*Starting with 108, I want to think of this one, I think about a cubed number.*0539

*108 is the same as 4 × 27.*0553

*That is important because 27 is one of my cubic numbers.*0558

*I will split it up over the 4 and 27.*0565

*We can go ahead and simplify that 3rd root of 27.*0570

*That one will be just 3.*0577

*We like to put our numbers first, let me write this one as 3 × 3rd root of 4*0581

*Note how we do not simplify the 4 because it is not a cubic number, it is a square number.*0589

*Let us try the next one, the 4th root of 160.*0597

*I want to think of numbers raised to the fourth power.*0603

*If you want you can even make a list.*0606

*1 ^{4} = 1, 2^{4} = 16, this is 4 16 × 10, the 4th root of 16, 4th root of 10.*0607

*We will go ahead and simplify the 4th root of 16 is 2, the 4th root of 10*0627

*That one is done.*0633

*Let us use the quotient rule on this last problem.*0640

*On top I will have the 4th root of 16 and in the bottom 4th root of 625*0644

*We saw earlier that the 4th root of 16 we will go ahead and simplify that will be just 2.*0652

*With 625 that is 5 ^{4}, 2/5.*0658

*Let us do the same thing with higher roots but we will deal with some variables underneath the roots.*0671

*The first one I have 3rd root of z ^{9}.*0679

*I could look at this as (z ^{3})^{3}.*0685

*What I want to do in this is highlight this root and that 3rd root end up getting rid of each other.*0693

*What is left over is z ^{3}.*0700

*Note that this meshes with some of our earlier work and I could write this as z ^{9}/3 and I will still get z^{3}.*0705

*All of our rules are staying nice and consistent with one another.*0715

*In here the3rd root of 8x ^{6}.*0719

*I’m looking at 3rd root of 8 and 3rd root of x ^{6}.*0724

*That will reduce to 2 and this would be like (x ^{2})^{3}.*0730

*2 × x ^{2} that will be the final reduced expression.*0745

*I have the 3rd root of 54 t ^{5}*0753

*In this one I will try to break down as much as possible but remember if we have things that are still not cubic,*0757

*we have to leave them underneath the root.*0763

*Let us see.*0766

*What cubic numbers can I find in 54?*0767

*That will be the same as 27 ×2 and what cubic numbers can I grab from t ^{5}?*0772

*That is a t ^{3} and t^{2}.*0785

*Notice here I am thinking of my product rule for exponents and how I have to add those exponents together to get 5.*0790

*I’m splitting them up just this way so that I would have one of them as cubed.*0797

*We could take the cubed of everything in here.*0805

*Cube of 27, 2 and t ^{3}, t^{2}*0808

*Some of these will simplify and some of them would not.*0817

*3rd root of 27 is 3, 3rd root of 2 has to stay, 3rd root of t ^{3} is t, and the 3rd root of t^{2} has to stay.*0821

*Gathering up what was able to be taken out I will have 3t × 3rd root of 2t ^{2}*0837

*Simplify and bring out as much as you can, and if you cannot go ahead and leave them under the root.*0846

*One more, I will first split up the root over the top and on the bottom.*0853

*A ^{15} / 3rd root of 64*0860

*On the top of this that is (a ^{5})^{3} underneath the cube root.*0868

*For the 64 on the bottom that is something that I could have just take the cube root of.*0879

*This would simplify into a ^{5} and 3rd root of 64 is 4.*0888

*That one is simplified.*0895

*Be familiar with your rules for these radicals especially when it gets to being able to split things up*0900

*and simplify over each of the individual components.*0907

*Thank you for watching www.educator.com.*0910

## Start Learning Now

Our free lessons will get you started (Adobe Flash

Sign up for Educator.com^{®}required).Get immediate access to our entire library.

## Membership Overview

Unlimited access to our entire library of courses.Learn at your own pace... anytime, anywhere!