WEBVTT mathematics/algebra-1/smith
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Welcome back to www.educator.com.
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In this lesson we are going to take a look at some complex fraction.
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We will first have to do a little bit about explaining what a complex fraction is
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and then I’m going to show you two techniques and how you can take care of them.
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In the first one we will write out these complex fractions as a division problem
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and then we will go ahead and use the method of this common denominator in order to simplify them.
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Each of these methods has their own advantages but they both should work when dealing with simplifying complex fraction.
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What a complex fraction is, it is either the numerator or the denominator is also a fraction.
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Here is a good example of a complex fraction using numbers.
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You will notice that the main division bars is actually sitting right here.
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But in the numerator I have 2/5 and in the denominator I have 1/7.
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If I have fractions made up of other fractions.
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These are exactly the types of things that we are looking to simplify.
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Since we are on a lot of rational expressions, then we will not only look at just numbers but we will look at more complex fractions like this.
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One of the first techniques that we can use to clean this up is to use division.
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We will use that main division bar and write this as a division problem.
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Here I have (x + ½) ÷ (6x + 3)/4x.
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It is the main division bar right there that will turn into division.
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That means we will have to use all of our tools for simplifying the left and right, and eventually be able to get them together.
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In a previous lesson we learn that we need to flip the second rational expression then multiply across the top and bottom.
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That is exactly what you will see with these.
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Let us grab on these rational expressions and give it a try.
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You have (t² u³) / r ÷ t^4u/r².
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We want to identify what is on the top and what is on the bottom?
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We want to write those again as a division problem.
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(t² u³) / r ÷ t^4u/r²
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It looks pretty good.
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We want to turn this into a multiplication process by flipping that second rational expression t² u³/ r will now be multiplied by r² ÷ t^4u.
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That looks much better.
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We can go ahead and multiply across the top.
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I will just put all of these on the top and multiply across the bottom.
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I will put of all of these on the bottom.
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Now that we have this we would simply go through and cancel out our common factors.
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We will get rid of t² on the top and t² in the bottom.
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Making a t² in the bottom.
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We can cancel out u, bring this down to u² and we can cancel out on r.
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We have r in the top.
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Let us write down everything that is left over.
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u² r ÷ t² and then we could consider this one simplified.
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It changes it into a problem that we have seen before.
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You just have to do a lot of work with simplifying.
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When it gets to simplifying a complex fraction, even that process is not necessarily the easiest to go through.
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In fact, you will find that this next problem is quite lengthy.
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We have (1/x + 1 + 2/y – 2) ÷ (2 /y – 2 – 1/x + 3).
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Let us identify everything on the top and everything in the bottom.
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That way we can simply rewrite this 1/x + 1 + 2/y – 2 all of this is being divided by everything on the bottom 2/y – 2 – 1/x + 3.
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If I have any hope I'm doing this as a division problem that I need to normally flip that second fraction.
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Notice how in this one I do not have a single fraction.
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It is tempting to say, hey why we just flip both of them but that is not how division works.
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We need to combine it into a single fraction before we can flip it and then do the multiplication.
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Let us see if we can get these guys together with some common denominators.
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The common denominators on this side are the x + 1 and y – 2.
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In order to get those together, I would have to give this fraction, y - 2 on the top and bottom.
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To make it workout over here we will give the top and bottom of that one x +1 and the 2 is still up there.
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It looks like that first piece will turn into y – 2 + let us go ahead and distribute this guy in there (2x + 2) ÷ (x + 1) (y – 2).
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That just takes that and crunches it down a little bit.
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Let us focus on this other one.
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We need a common denominator and I see there is a y - 2 and x + 3.
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I will need to give the fraction on the left an additional x + 3.
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We still have the 2 in there.
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2 × x + 3
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Over on the other side let us give the top and bottom of that one y -2.
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When those are put together, we will do a little bit of distributing here.
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We will have (2x + 6) – (y + 2) ÷ (y – 2) (x + 3).
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Let us go ahead and write this again and see if we can do the actual division.
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It looks like I can cancel out a few things in here.
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Let us save ourselves a little bit of work.
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y + 2x ÷ (x + 1) (y -2), we are dividing it by the second fraction.
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Here is when I’m going to flip and multiply it, multiplied by 2x - y + 8.
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I have combined the 6 and the 2 together, y - 2 x + 3.
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I can just combine the tops and bottoms.
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Quite a lot of pieces in here that is okay, at least I see one piece will cancel out and that is the y – 2.
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We are left with y + 2x and (x + 3) ÷ (x +1) and 2x - y + 8.
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Even though it can be a lengthy process, by rewriting it as a division process and using our tools from before
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you will see is that it is possible to reduce and simplify this complex fractions.
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The other method which can often be a lot cleaner is using the least common denominator
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in order to clear out all the fractions in the top and bottom.
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In order for this method to work, you must find the least common denominator of all the little fractions present in your complex fraction.
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Go ahead and look at your numerator and denominator and think of all the least common denominator for all those fractions.
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Once you find that LCD, then you are going to multiply on the top and the bottom of the main fraction
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or on the top and bottom of the main division bar.
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This will clear up things immensely but you have to be careful on canceling out.
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You do not want to accidentally cancel out something that you should not.
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You will see that you will clear out a bunch of stuff and then you end up simplifying just as you would normally.
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No matter what method you use, you should get the same answer.
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Use the method that you are more comfortable with.
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I like to recommend a method two because it is usually much cleaner than using the first method.
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However, anything that is cleaner use less opportunities for mistakes.
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One downside to the second method is usually happen so quickly it is hard to keep track of everything that was in there.
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Let us try this second method with the following complex fraction.
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I have (2/s² t + 3 /st²) ÷ (4/s² t² – 1/st).
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Let us first see if we can identify the least common denominator.
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Try and pick out all these little denominators here, see what the LCD would be.
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They all have some s and the largest one I see there is an s².
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They all have t’s but the largest one in there is a t².
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I’m going to take this and I’m going to multiply it on the top and the bottom of this expression so I can rewrite it.
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There is our main division bar.
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Let us get to multiplying.
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On the top of this entire main division bar I will multiply it by this LCD.
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I will do the same thing on the bottom to keep things nice and balanced.
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The top and bottom both have two terms, so we will definitely make sure that we distribute each of these parts.
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We are going to write the result of this multiplication and then watch how many things will cancel out in the next step.
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(2s² t² / s² t) + (3s² t² / st²) ÷ (4s² t / s² t² - s² t/st).
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It looks messy and it looks I have actually made things even more complicated.
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Watch what is going to go take a vacation here soon.
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I have these s² they will go away.
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Then I have a single t and t up top, they will go away.
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On the next one, there is my s one of those will be gone and both my t² are gone.
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Onto the bottom s² is gone, s² is gone.
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I think I am missing some of my squares.
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Our t² are gone and we can get rid of one of these t’s.
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The denominators of all those little fractions, these guys that we are so worried about at the very beginning,
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all of them have been cancel out in some way or another.
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This means as soon as we write down our leftover pieces, this one is simplified.
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2t + 3s, those are the only things that survived up here ÷ 4 - st and those are the only things that survived on the bottom.
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This one is in its most simplified form.
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It is nice, quick and easy method you just have to properly identify the LCD first.
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You will know you are using the method right if all of these denominators end up going away.
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If any of them are still in there double check to see what LCD you used.
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Let us try this one more time but something a little bit more complicated.
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This one is (2y + 3 / y – 4) ÷ (4y² – 9) + (y² – 16).
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Let us examine these denominators so we can find our LCD.
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We are looking at this one this is the same as y - 4 and y + 4.
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Over here it already has the y -4 in it.
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The only piece that I am missing is the y + 4.
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My LCD will contain both of these parts.
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I have the y – 4 and y + 4, both of those in there.
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I’m going to take that and we are going to multiply it on the top and bottom of the original.
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Let me just quickly rewrite this and I’m going to rewrite it with the factored form on the bottom.
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We will take our LCD and we will multiply it on the top and on the bottom of our main fraction here.
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y – 4 y + 4.
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Let us go ahead and put everything together and let us see how this looks.
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I have (2y + 3 ) (y – 4) (y + 4) ÷ y – 4.
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Then comes our main division bar right there.
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On the bottom is (4y² – 9) × (y – 4) × (y + 4) ÷ (y – 4) (y + 4).
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Watch how many things will cancel in this next step.
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y – 4 and y – 4 those are gone.
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y – 4, y – 4, y + 4, y + 4, 4 those are gone.
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All of these problems that we had at the very beginning, they are no longer problem.
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They are gone.
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We will simply write down all of the left over pieces.
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2y + 3 y + 4 4y² – 9.
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Be careful, there is still some additional reducing that you can do even after using your LCD like this.
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One thing that I can see is that I can actually continue factoring the bottom.
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Let us write that out.
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2y + 3/ y + 4 and this will be over I have different squares on the bottom so, 2y + 3, 2y – 3.
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I’m sure enough now we can more easily that I have an extra 2y – 3 in the bottom and that is gone as well.
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This one finally reduces down to y + 4 / 2y -3 and now we are finally done.
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The second method is definitely handy and clears up a lot of fractions very quickly.
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Use whatever method you are more comfortable with.
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