WEBVTT mathematics/algebra-1/smith
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Welcome back to www.educator.com.
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In this lesson let us take a look at how you can multiply and divide radicals.
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We will first cover some rules for multiplying and dividing radicals and then get into that division process
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and see how we want to rationalize the denominator that involves a radical expression.
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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.
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For example, you can change radicals into fractional exponents.
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If you want to combine them for division you can separate those out if you want to separate them out over multiplication.
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You can also add and subtract radicals as long as we make sure that the radicands and the indexes are exactly the same.
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Radicals follow our properties for all the other types of numbers so we can also use
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the commutative property, associative property, even on these radicals.
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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.
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Treat these radicals like they are any other numbers.
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Let us see this as we walked through the following problem.
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I want to multiply the √6 + 2 × √6-3
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For this one treat it like binomials and use foil to multiply everything out.
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Every terms multiplied by every other term.
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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.
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Then we can use our other rules to go ahead and simplify this further.
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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.
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-3 × √6 + 2 × √6 can I add them together or not?
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Yes I can because they have the same radical I will just do the coefficients.
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-3 + 2 - √6
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- 6 is still there.
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You can continue simplifying your like terms by combining the 6 and -6 giving -√6.
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We are using many of our tools that we have learned up to this point, in order to handle these radicals.
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Watch out for lots of situations where you need to use foil.
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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.
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However, do not do that.
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Do not even attempt any type of distribution with this one.
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What you should do with it is foil because as we learned in our exponent section, this stands for 4 + √7 and 4 + √7 .
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Those two things are being multiplied by each other.
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I can see where foil comes in to play.
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I will take my first terms 16, outside terms, inside terms, and last terms.
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Unlike before I could use some other rules to go ahead and combine things.
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I can go ahead and add these two since they have the same radical and get 8√7.
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I can combine these under the same radical and get the √49 and that is simply 7.
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We can go ahead and finish off this problem, 16 + 7 would be 23 or 23 + 8 × √7.
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Let us take a close look at division.
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When dividing a radical expression we go ahead and rewrite it by rationalizing the denominator.
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If you have never heard that term before rationalizing the denominator,
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it is a way of rewriting it so that there is no longer a radical in the bottom.
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That seems a little weird.
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I mean, if we are interested in dividing by radical why are we writing it that there is no radical in the bottom.
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It seems like we are side stepping the problem like we are not ending up dividing by the radical.
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It is just a simpler way of looking at the whole division process and it is something that you have done before with fractions.
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For example, when we have 1/3 ÷ 2/5 you have been taught that you should always flip the second and then multiply.
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Why are we doing that? Why do not we just go ahead and divide by 2/5?
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What is the big deal with a turn it into multiplication problem?
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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.
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Also supposed I write this problem as 1/3 ÷ 2/5, you recognize that this is one of our complex fractions.
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I can simplify a complex fraction by multiplying the top and bottom by the same thing.
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I will multiply the top and bottom by 5/2, that should be able to get rid of our common denominator.
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On top I would have 1/3 × 5/2 and on the bottoms 2/5 × 5/2 would be 1.
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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.
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That we should take the second one, flip it and multiply.
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But notice how we are doing that by simplifying the bottom now we are dividing by 1.
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It is a great way to end up rewriting the problem, and taking care of it in a much simpler way.
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That is exactly what we want to do when we are rationalizing.
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The actual steps for rationalizing the denominator look a lot like this.
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First, we will end up rewriting the rational expression, so that we will end up with no root in the bottom.
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When we rationalize we try to get rid of that root.
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We will do this by multiplying the top and bottom by the smallest number that gets rid of that radical expression.
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That sometimes you can use some larger numbers, but it is best to use the smallest thing that will get rid of that radical.
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It saves you from doing some extra simplifying in the end.
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If we are dealing with square roots I recommend try and make that perfect square
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in the denominator and that should be able to rationalize it just fine.
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Let us see one of this division by radical in process and this is also known as rationalizing the denominator.
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I have 2 ÷ √(2 )
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I’m going to end up rewriting this so that there is no longer √2 in the bottom.
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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.
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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.
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I have created that square number on the bottom and now we can go ahead and simplify it.
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That will be 2.
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If you can simplify from there go ahead and do so, you will see that this one turns into the √2.
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When I take 2 ÷√2 like the original problem says the √2 is my answer.
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Let us try another one.
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This one is 12 ÷ √5
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We are looking to multiply the top and bottom by something to get through that √5 in the bottom we will use another√5.
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12 √5 for the top and bottom √5 × √5 would be the same as √25.
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The good news is that one reduces and becomes just 5.
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If you are doing a division, you are dividing by the √5 even though it looks like you are changing into multiplication problem.
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This is also known as rationalizing the denominator.
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You can use these tools like rationalizing the denominator for some much higher roots as well.
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The thing to remember by is that you want to multiply by the smallest root actually complete set root.
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When we are dealing with the square roots it look like always multiplying by the same thing on the bottom.
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With high roots, sometimes that might not be the case.
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Let us look at this one.
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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.
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We will be left with the 3rd root of 2 / 3rd root of 4 and 4 is not a cubic number.
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It is like it did not have enough of the number on the bottom to go ahead and simplify it completely.
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Which should we multiply?
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If I want a cubic number in the bottom but I'm going to use 3rd root of 4 .
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When I take that onto the top and bottom you will see that indeed we do get that cubic number that we need.
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From the bottom this would be 3rd root of 8 and on the top I will just have 3rd root of 4 .
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The bottom simplifies becomes 2 now my answer is 3rd root of 4 ÷ 2.
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Let us try another one of those higher roots.
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This one is 3 ÷ 4th root of 9
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Let us see, what would I have to use with a 4th root of 9?
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9 is the same as 3 and 3, it will be nice if I had even more 3’s underneath there.
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Let us say a couple of more.
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I will accomplish that by doing the top and bottom by 4th root of 9 .
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3 × 4th root of 9 for the top and 4th root of 81 on the bottom.
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Now it is looking much better.
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34th root of 9 on the top, the bottom simplifies to just the 3 and now we will cancel out these extra 3s.
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I have 4 4th root of 9 .
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This one we want to rationalize the denominator and if you look at it you must think what denominator are we trying to rationalize?
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Is there a root in the bottom?
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Because of our rules that allow us to break up the root over the top and bottom, there is.
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In this one we have the 3rd root of 2y / 3rd root of z.
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We can see we have 3rd root of z and it definitely needs to be simplified.
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How are we going to do that?
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Since I'm dealing with a cubed root I will need an additional 2 z's for that bottom.
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Let us use the cube root of z².
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Watch what that would give us here on the bottom.
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3rd root of z³ on top, the 3rd root of 2y z².
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The bottom simplifies and there will no longer be any roots in the bottom.
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3rd root of 2y z² / z and I know that this one is done.
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One thing that can make the rationalization a more difficult process since we have more than one term in the bottom.
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Our main goal is to end up rewriting the bottom so that there is no longer a root.
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If we have more than one term we are going to end up using something known as
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the conjugate of the expression to go ahead and get rid of it.
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What the conjugate is, it is the same as our original expression, but it has a different sign connecting them.
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That will allow us to get rid of that root.
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To show you why we get through the root we will use an example.
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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.
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Watch what happens when I foil these two together.
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The 4 + √3 and its conjugate.
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Starting with the first terms I have 4 × 4, which would be 16.
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My outside terms would be -4√3, my inside terms will be 4√3.
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I can move on to my last terms -√3 × √3.
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A lot of things are happening when you multiply by its conjugate.
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One, notice how our outside and inside terms where the same but one was positive and one was negative.
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When you are dealing with conjugates that should always happen.
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Those two things are gone.
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We will focus was going on down here on the end.
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-√3 × √3, -3 × 3 which be 16 - √9 which would be 9.
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The numbers may change to make it different but at this point, there is no more radical.
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Because those two radicals multiply and I get that perfect square number, there is no more radicals to deal with.
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I can just take 16 - 9 and get a result of 7.
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By multiplying by that conjugate, I got rid of all instances of all radicals.
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This is why it will be important to use it when getting rid of our radical on the bottom.
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Let us see for these examples.
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Notice we need that conjugate because we have two terms in the denominator.
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I'm going to multiply the top and bottom of this one by the conjugate.
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√5 – 2 and √5 - 2
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When dealing with more than one term, remember that you will have to foil out the bottom.
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Also remember that though you will have to possibly foil or even distribute the top.
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The top 9√5 – 18 and working with the bottom and foiling that out my first terms would be √5 × √5 =5.
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Outside terms and inside terms they are going to cancel out I know I’m on the right track.
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2 × -2 -4
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We will go ahead and do some canceling and let us see what we have left over.
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9√5 - 18 / 5 – 4 = 1 and this reduces to 9√5 - 18.
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Notice how we have divided by that radical because we have gotten rid of that radical in the bottom.
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In this last example, let us look at rationalizing the following denominator.
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We have 7 ÷ 3 - √x
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With this one I’m going to have to multiply the top and the bottom by its conjugate.
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You know what is next in there, this will be 3 + √x we will do that on the bottom and on the top.
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Let us see what this does to the top as we distribute and remember that on the bottom we will foil.
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I get 21 + 7√x for the top.
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On the bottom we have 9 + 3 × √x - 3 × √x and then - √x × √x.
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If we do things correctly we should get rid of all those radicals in the bottom.
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+ square root - square root those two will take care of each other and then my √x× √x = x.
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This will leave us with 21 + 7 √x / 9 – x.
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We have got rid of all those radicals in the bottom you can say that our denominator is rationalized.
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You can see that when you are working with dividing radicals you always have to keep in mind
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what you would put on the bottom in order to get rid of all those radicals.
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If you only have a single term feel free to multiply by what would complete whatever that radical is.
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Complete the square or complete the cube.
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If you have more than one term use the conjugate to go ahead and rationalize the denominator.
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