LCD Method 1 Part 2-rational expressions

Uploaded by videosbyjulieharland on 06.05.2010

>> In this video, I'm going to go over how
to find the least common denominator
of algebraic fractions.
These will be a little bit more advanced,
where we've got three different fractions.
And, so we're just stepping up the pace a little bit.
Remember, we use LCD as a little abbreviation
for least common denominator.
And the LCD is simply the least common multiple
of the denominators.
So, what we're doing is focusing on the denominators.
The reason we want to find the LCD is because it's
in preparation for adding fractions.
When we have different denominators, we need to figure
out how to get a common denominator
in order to add the fractions.
So, for numbers, I went over several ways
of getting the least common multiple.
This is on my videos on fractions.
So if you want to know how to do it for numbers, look there.
Let's say you had 5 over 9x to the 4th y and 7 over 15xy cubed,
and 8 over, let's say 10x squared y to the 4th.
Okay, and I want to know what the least common multiple
of the denominators is.
One way of doing this is simply write everything
in prime factorization form, including the numbers.
So, if I did that, and I'm also going
to use exponents when necessary.
So, for instance, this one becomes 3 squared x
to the 4th times the y. This one becomes 3 times 5 times x times
y cubed.
And this one becomes 2 times 5 times x squared times y
to the 4th.
So, this method I'm going to show you
for getting the least common multiple is to write everything
in prime factor form, but if there's the same factor more
than once, then write it using exponents.
Now, what we're going to do is look
at the different types of factors.
For instance, 2 is a different factor than 3,
x is a different factor than y. And just write them all
down multiplied together.
So, there's a 2, there's a 3, there's an x
and there's a y. There are no other different kinds
of factors.
So, this is going to be the least common multiple
of the denominators.
And what you do is you choose the exponent -- oops.
I forgot a 5, just forgot that 5.
So there we go.
We've got a 2, a 3, a 5 an x and a y. Then you see if any
of these have exponents, you choose the highest exponent.
All right.
there's only one place where there's a 2,
and there's no exponents.
There's two places where there's a 3 and there's a 3 squared.
So you need to really to have a 3 squared.
Then you look for the 5.
There's no exponents on any of the 5's.
For the x's, the highest exponent would be this one,
x to the 4th.
And for the y's, the highest exponent is the 4th.
And so, there is the least common denominator
of these three fractions.
So, basically, this was the least common multiple
of these three denominators, okay?
So, that's one way of doing it.
You simply write the prime factorization using exponents
if you've got a factor more than once.
Let's do another one.
So, here's another problem.
What we need to do is factor the denominator.
So, we're looking for the least common denominator
of all three rational expressions or fractions.
So, for the first one, the way to factor 5x plus 10 is to take
that 5, that's the greatest common factor.
Now, for 75, notice I have not got this written
in prime factored form.
So you could use a factor tree
or anything you want to write that.
But, see, I think of that as 3 times 25,
so I have a 3 times a 5 squared and I have an x squared.
And for the last one, I have a difference of two squares,
so that's x plus 2 times x minus 2.
All right.
So what we've done is prime factored all of them.
Now we have to look for the different kinds of factors.
So, I'm going to write the least common denominator.
Yeah, I kind of use it interchangeably in a way.
Now I'm thinking of the least common denominator
of these three fractions.
So, what are the different kinds of factors?
Well, I have like a 5, right?
I've got a 3.
I've got an x. So, first of all, I look for the, you know,
the single monomial type factors.
And then I have also an x plus 2 and I have an x minus 2.
All right.
So, for the 5, you look and see anywhere there's a 5,
you pick any that have the highest exponent.
That'll be a 5 squared from right here, right?
We're going to choose the 5 squared.
Okay, for the 3, there's only one 3, and so this is the 3.
For the x, there's only one place where there's an x,
right here, all right?
And also that has the 2 on it, so the x square.
All right, x plus 2, hmm?
There aren't any exponents on x plus 2.
And x minus 2, no exponents.
So, all together, this is your least common denominator.
Now, I usually multiply the numbers back out so
that 25 times 3, 75x squared.
And you could either write x plus 2 times x minus 2
or x squared minus 4.
It depends on how you want to finish writing it.
Now, as far as the numbers go, once you factor,
if you left this as 75 instead of 3 times 5 squared,
for the numbers, you could figure out the numerical part,
the 75 some other way as well
without doing this prime factorization.
I'm just doing all of them the same exact way.
All right.
Here's another example.
And keep in mind, on all of these, I'm not paying attention
at all to the numerator.
All I'm trying to decide is what the least common denominator
of the three algebraic fractions are.
And you might -- usually you only have two
that you might be adding together.
That would be even easier.
All right, so all we're doing is factoring the denominators,
All right.
So, for this first one, x squared plus 6x plus 9,
hopefully you recognize that that is a perfect square.
If you don't you might do it the long way
and get x plus 3 times x plus 3.
But using this method I've shown you, we want to write
that with the exponent.
So, it's x plus 3 squared.
All right, next one.
We take out the greatest common factor.
So that will be 5x to the 4th, all right?
And that gives you x plus 3.
And the last one, we've got 10x --
now I could leave that as 10x,
or I could write that as 2 times 5x.
[ Pause in Lecture ]
All right.
So, now we're going to look at the different kinds of factors.
Well, I've got an x plus 3 and I have a 5 as a factor.
I have an x as a factor.
Let's see -- and I have a 2 as a factor.
And I think those are the different types of factors.
So, I'm not looking at any of the exponents yet.
Now, I look back up and I take the highest exponent
that appears on any of these.
So, for x plus 3, 2 is the one --
whatever denominator has the highest exponent.
5 is only -- there's none that have any exponents on that,
which means, of course, that's an exponent of 1.
For an x, I've got an x to the 4th over here and an x. So,
I want the x to the 4th, highest exponent.
And for 2, there's only one of them,
and it's 2 to the 1st power.
So, it's usually easier to write this as 10x
to the 4th, x plus 3 squared.
Now, these are harder than most problems that you're going
to encounter when you're adding fractions.
But this is how you could get the least common denominator.
And keep in mind that if I did leave this as 10x,
you could just decide that the least common denominator of 5
and 10 is just 10, all right?
So you might not have done it in the factored form.
So, you could either write that as 2 times 5 times x, right?
Or, you could write that as just 10x.
[ Pause in Lecture ]
All right.
Let's do one more.
So, we're going to start off by factoring each.
This is a 3 times x plus 8.
The second one, denominator is a difference of two squares,
x plus 8 times x minus 8.
And the third one, you could take out a common factor
of 9, which is an x plus 8.
All right.
So, let's see what we have here
for the least common denominator.
We're looking for -- well, let's see.
What kind of factors do I have?
How about if we just look at the number ones, you know,
just the coefficients.
I have a 3 and a 9.
You could write that 9 as 3 squared.
Notice I left it as 9.
But what's the least common multiple of 3 and 9?
Hopefully you know that's just 9,
which you would have also gotten
if you'd written this as 3 squared, right?
Because you would take the highest exponent of 3 squared.
And let's see.
I have an x plus 8.
And if you look throughout all of them, there's no exponent
on any that have an x plus 8.
And the other kind of factor is an x minus 8.
So, that's actually the answer to this one,
because again x minus 8 doesn't have an exponent.
Now, you could leave your answer like that,
or you could multiply it
out by writing 9 times x squared minus 64.
And then, if you want, you could also multiply that out by 9.
So, there's different ways
of writing the least common multiple.
All right.
So these are some problems we've done giving the least common
multiple of the denominators
for three fractions, algebraic fractions.