Algebra 1 Multiply & Divide Radicals

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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 process0011

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 use0059

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 × √6-30087

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 - √60147

- 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 square0434

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 ÷ √50515

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 90654

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 z².0758

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

3rd root of z³ on top, the 3rd root of 2y z².0768

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

3rd root of 2y z² / 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 as0812

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 - 20968

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 -41015

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 - √x1050

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 mind1139

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

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