### Multiply & Divide Radicals

- When multiplying radical expressions of the same index we can combine them into a single radical.
- Watch for situations that we need to use FOIL in the multiplication process. These situations involve two binomials.
- To divide by a radical with rationalize its denominator. This means we multiply the top and bottom by an expression that completes the rational in the denominator. There should no longer be any radicals in the denominator.
- When dividing by a binomial that has a rational expression use the conjugate of the expression. The conjugate is almost the same, but is connected by a different sign.

### Multiply & Divide Radicals

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 0:08
- Multiply and Divide Radicals 0:25
- Rules for Working With Radicals
- Using FOIL for Radicals
- Don’t Distribute Powers
- Dividing Radical Expressions
- Rationalizing Denominators
- Example 1 7:22
- Example 2 8:32
- Multiply and Divide Radicals Cont. 9:23
- Rationalizing Denominators with Higher Roots
- Example 3 10:51
- Example 4 11:53
- Multiply and Divide Radicals Cont. 13:13
- Rationalizing Denominators with Conjugates
- Example 5 15:52
- Example 6 17:25

### Algebra 1 Online Course

### Transcription: Multiply & Divide Radicals

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

*In this lesson let us take a look at how you can multiply and divide radicals.*0003

*We will first cover some rules for multiplying and dividing radicals and then get into that division process*0011

*and see how we want to rationalize the denominator that involves a radical expression.*0018

*There are only a few rules that you have to remember when working with radicals and the good news is we have seen many of these already.*0027

*For example, you can change radicals into fractional exponents.*0033

*If you want to combine them for division you can separate those out if you want to separate them out over multiplication.*0039

*You can also add and subtract radicals as long as we make sure that the radicands and the indexes are exactly the same.*0050

*Radicals follow our properties for all the other types of numbers so we can also use*0059

*the commutative property, associative property, even on these radicals.*0064

*The reason why that is important is because there is a few situations where you often try to apply some rules that do not work.*0072

*Treat these radicals like they are any other numbers.*0080

*Let us see this as we walked through the following problem.*0084

*I want to multiply the √6 + 2 × √6-3*0087

*For this one treat it like binomials and use foil to multiply everything out.*0093

*Every terms multiplied by every other term.*0099

*Let us see our first terms would have √6 × √6, outside terms -3 √6, inside terms 2√6 and our last terms 2 × -3 = 6.*0103

*Then we can use our other rules to go ahead and simplify this further.*0119

*I see I have two roots here, I can put those underneath the same root and that is the same as the √36 or 6.*0125

*-3 × √6 + 2 × √6 can I add them together or not?*0135

*Yes I can because they have the same radical I will just do the coefficients.*0142

*-3 + 2 - √6*0147

*- 6 is still there.*0157

*You can continue simplifying your like terms by combining the 6 and -6 giving -√6.*0159

*We are using many of our tools that we have learned up to this point, in order to handle these radicals.*0168

*Watch out for lots of situations where you need to use foil.*0176

*In this example I have 4 + √7 ^{2} it is tempting to try and take that 2 and distribute it over the parts in between.*0181

*However, do not do that.*0191

*Do not even attempt any type of distribution with this one.*0194

*What you should do with it is foil because as we learned in our exponent section, this stands for 4 + √7 and 4 + √7 .*0197

*Those two things are being multiplied by each other.*0208

*I can see where foil comes in to play.*0212

*I will take my first terms 16, outside terms, inside terms, and last terms.*0216

*Unlike before I could use some other rules to go ahead and combine things.*0227

*I can go ahead and add these two since they have the same radical and get 8√7.*0232

*I can combine these under the same radical and get the √49 and that is simply 7.*0239

*We can go ahead and finish off this problem, 16 + 7 would be 23 or 23 + 8 × √7.*0249

*Let us take a close look at division.*0266

*When dividing a radical expression we go ahead and rewrite it by rationalizing the denominator.*0269

*If you have never heard that term before rationalizing the denominator,*0278

*it is a way of rewriting it so that there is no longer a radical in the bottom.*0281

*That seems a little weird.*0286

*I mean, if we are interested in dividing by radical why are we writing it that there is no radical in the bottom.*0287

*It seems like we are side stepping the problem like we are not ending up dividing by the radical.*0294

*It is just a simpler way of looking at the whole division process and it is something that you have done before with fractions.*0299

*For example, when we have 1/3 ÷ 2/5 you have been taught that you should always flip the second and then multiply.*0307

*Why are we doing that? Why do not we just go ahead and divide by 2/5?*0314

*What is the big deal with a turn it into multiplication problem?*0317

*One, we will show you how accomplish the same thing, but by flipping the second one multiplying it does it in a much simpler way.*0321

*Also supposed I write this problem as 1/3 ÷ 2/5, you recognize that this is one of our complex fractions.*0330

*I can simplify a complex fraction by multiplying the top and bottom by the same thing.*0341

*I will multiply the top and bottom by 5/2, that should be able to get rid of our common denominator.*0348

*On top I would have 1/3 × 5/2 and on the bottoms 2/5 × 5/2 would be 1.*0359

*What we are sitting on the top there is the 1/3 and there is the 5/2 which comes from that rule we learned.*0369

*That we should take the second one, flip it and multiply.*0379

*But notice how we are doing that by simplifying the bottom now we are dividing by 1.*0382

*It is a great way to end up rewriting the problem, and taking care of it in a much simpler way.*0390

*That is exactly what we want to do when we are rationalizing.*0396

*The actual steps for rationalizing the denominator look a lot like this.*0402

*First, we will end up rewriting the rational expression, so that we will end up with no root in the bottom.*0407

*When we rationalize we try to get rid of that root.*0413

*We will do this by multiplying the top and bottom by the smallest number that gets rid of that radical expression.*0416

*That sometimes you can use some larger numbers, but it is best to use the smallest thing that will get rid of that radical.*0422

*It saves you from doing some extra simplifying in the end.*0429

*If we are dealing with square roots I recommend try and make that perfect square*0434

*in the denominator and that should be able to rationalize it just fine.*0437

*Let us see one of this division by radical in process and this is also known as rationalizing the denominator.*0444

*I have 2 ÷ √(2 )*0451

*I’m going to end up rewriting this so that there is no longer √2 in the bottom.*0456

*We will do this by multiplying the top and bottom by another root so that we will have a squared number in the root for the bottom.*0462

*On top I have 2√(2 ) and in the bottom I can go ahead and put these together and get √2 × 2, which is the √4.*0471

*I have created that square number on the bottom and now we can go ahead and simplify it.*0485

*That will be 2.*0493

*If you can simplify from there go ahead and do so, you will see that this one turns into the √2.*0496

*When I take 2 ÷√2 like the original problem says the √2 is my answer.*0503

*Let us try another one.*0513

*This one is 12 ÷ √5*0515

*We are looking to multiply the top and bottom by something to get through that √5 in the bottom we will use another√5.*0519

*12 √5 for the top and bottom √5 × √5 would be the same as √25.*0529

*The good news is that one reduces and becomes just 5.*0541

*If you are doing a division, you are dividing by the √5 even though it looks like you are changing into multiplication problem.*0549

*This is also known as rationalizing the denominator.*0558

*You can use these tools like rationalizing the denominator for some much higher roots as well.*0565

*The thing to remember by is that you want to multiply by the smallest root actually complete set root.*0571

*When we are dealing with the square roots it look like always multiplying by the same thing on the bottom.*0577

*With high roots, sometimes that might not be the case.*0583

*Let us look at this one.*0586

*1 / 3rd root of 2 if I try and multiply the top and bottom by 3rd root of 2 it is not going to get rid of that radical.*0587

*We will be left with the 3rd root of 2 / 3rd root of 4 and 4 is not a cubic number.*0598

*It is like it did not have enough of the number on the bottom to go ahead and simplify it completely.*0604

*Which should we multiply?*0612

*If I want a cubic number in the bottom but I'm going to use 3rd root of 4 .*0615

*When I take that onto the top and bottom you will see that indeed we do get that cubic number that we need.*0625

*From the bottom this would be 3rd root of 8 and on the top I will just have 3rd root of 4 .*0632

*The bottom simplifies becomes 2 now my answer is 3rd root of 4 ÷ 2.*0640

*Let us try another one of those higher roots.*0652

*This one is 3 ÷ 4th root of 9*0654

*Let us see, what would I have to use with a 4th root of 9?*0659

*9 is the same as 3 and 3, it will be nice if I had even more 3’s underneath there.*0664

*Let us say a couple of more.*0671

*I will accomplish that by doing the top and bottom by 4th root of 9 .*0674

*3 × 4th root of 9 for the top and 4th root of 81 on the bottom.*0682

*Now it is looking much better.*0693

*34th root of 9 on the top, the bottom simplifies to just the 3 and now we will cancel out these extra 3s.*0696

*I have 4 4th root of 9 .*0706

*This one we want to rationalize the denominator and if you look at it you must think what denominator are we trying to rationalize?*0715

*Is there a root in the bottom?*0722

*Because of our rules that allow us to break up the root over the top and bottom, there is.*0725

*In this one we have the 3rd root of 2y / 3rd root of z.*0731

*We can see we have 3rd root of z and it definitely needs to be simplified.*0744

*How are we going to do that?*0750

*Since I'm dealing with a cubed root I will need an additional 2 z's for that bottom.*0752

*Let us use the cube root of z ^{2}.*0758

*Watch what that would give us here on the bottom.*0765

*3rd root of z ^{3} on top, the 3rd root of 2y z^{2}.*0768

*The bottom simplifies and there will no longer be any roots in the bottom.*0779

*3rd root of 2y z ^{2} / z and I know that this one is done.*0784

*One thing that can make the rationalization a more difficult process since we have more than one term in the bottom.*0797

*Our main goal is to end up rewriting the bottom so that there is no longer a root.*0805

*If we have more than one term we are going to end up using something known as*0812

*the conjugate of the expression to go ahead and get rid of it.*0815

*What the conjugate is, it is the same as our original expression, but it has a different sign connecting them.*0820

*That will allow us to get rid of that root.*0827

*To show you why we get through the root we will use an example.*0830

*Here I have 4 + √3, the conjugate of this one would be the same I have a 4√3 it will have a different sign connecting them 4 - √3.*0835

*Watch what happens when I foil these two together.*0852

*The 4 + √3 and its conjugate.*0855

*Starting with the first terms I have 4 × 4, which would be 16.*0863

*My outside terms would be -4√3, my inside terms will be 4√3.*0868

*I can move on to my last terms -√3 × √3.*0877

*A lot of things are happening when you multiply by its conjugate.*0882

*One, notice how our outside and inside terms where the same but one was positive and one was negative.*0886

*When you are dealing with conjugates that should always happen.*0892

*Those two things are gone.*0897

*We will focus was going on down here on the end.*0899

*-√3 × √3, -3 × 3 which be 16 - √9 which would be 9.*0903

*The numbers may change to make it different but at this point, there is no more radical.*0918

*Because those two radicals multiply and I get that perfect square number, there is no more radicals to deal with.*0925

*I can just take 16 - 9 and get a result of 7.*0932

*By multiplying by that conjugate, I got rid of all instances of all radicals.*0940

*This is why it will be important to use it when getting rid of our radical on the bottom.*0946

*Let us see for these examples.*0953

*Notice we need that conjugate because we have two terms in the denominator.*0955

*I'm going to multiply the top and bottom of this one by the conjugate.*0962

*√5 – 2 and √5 - 2*0968

*When dealing with more than one term, remember that you will have to foil out the bottom.*0976

*Also remember that though you will have to possibly foil or even distribute the top.*0984

*The top 9√5 – 18 and working with the bottom and foiling that out my first terms would be √5 × √5 =5.*0991

*Outside terms and inside terms they are going to cancel out I know I’m on the right track.*1006

*2 × -2 -4*1015

*We will go ahead and do some canceling and let us see what we have left over.*1020

*9√5 - 18 / 5 – 4 = 1 and this reduces to 9√5 - 18.*1024

*Notice how we have divided by that radical because we have gotten rid of that radical in the bottom.*1037

*In this last example, let us look at rationalizing the following denominator.*1046

*We have 7 ÷ 3 - √x*1050

*With this one I’m going to have to multiply the top and the bottom by its conjugate.*1055

*You know what is next in there, this will be 3 + √x we will do that on the bottom and on the top.*1064

*Let us see what this does to the top as we distribute and remember that on the bottom we will foil.*1073

*I get 21 + 7√x for the top.*1082

*On the bottom we have 9 + 3 × √x - 3 × √x and then - √x × √x.*1089

*If we do things correctly we should get rid of all those radicals in the bottom.*1108

*+ square root - square root those two will take care of each other and then my √x× √x = x.*1114

*This will leave us with 21 + 7 √x / 9 – x.*1123

*We have got rid of all those radicals in the bottom you can say that our denominator is rationalized.*1133

*You can see that when you are working with dividing radicals you always have to keep in mind*1139

*what you would put on the bottom in order to get rid of all those radicals.*1144

*If you only have a single term feel free to multiply by what would complete whatever that radical is.*1148

*Complete the square or complete the cube.*1154

*If you have more than one term use the conjugate to go ahead and rationalize the denominator.*1157

*Thank you for watching www.educator.com.*1162

1 answer

Last reply by: Rhonda Steed

Tue Apr 22, 2014 12:20 AM

Post by Deepa Kumar on April 11, 2014

at 15:20 you said 16 minus the square root of 9 equals 16 minus 9 and that equals 7 while the correct answer is 16 minus 3 which equals 13.

2 answers

Last reply by: Mohamed Elnaklawi

Sat Apr 19, 2014 9:56 AM

Post by Mohamed Elnaklawi on April 8, 2014

what does foil stand for?