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>> Okay, ready to do 7.3.
All righty.
All right, 7.3 deals with complex fractions.
And what I mean by complex fractions is a fraction
within a fraction.
So I have a fraction here, I have a fraction here,
and the whole thing is a fraction.
Now, what does it mean literally?
Don't look at it that way, look at it like this.
I means X plus 1 over X divided by X minus 1 over 2X.
Okay? We know -- oops, parentheses.
We know that when you divide you have to flip and multiply.
We also know that when you have something added
or subtracted the only way you can reduce it is
if you have something exactly the same.
There's nothing I can do to change either one of those,
right, factor farther,
so typically the only thing I can do to reduce would be this X
and that X. Which makes your answer X plus 1 times the 2
over X minus 1.
And that's as far as you can go.
But I took this complex factor and just wrote it
out literally what it means.
Another one.
2 plus 1 over Y divided by 3 minus 2
over Y. Again, literally this means.
2 plus 1 over Y divided by 3 minus 2 over Y. Now,
in this first one we had whole fraction, whole fraction.
We flipped and multiplied.
We don't have a whole fraction here
and we don't have a whole fraction here.
We have things that are being added or subtracted,
so we need to combine it to be one fraction,
which is what we just did in 7.2.
In other words, I have 2 over 1 plus 1 over Y that I need
to combine, and I have 3 over 1 minus 2 over Y
that I need to combine.
So what would the common denominator be
for this, 1 and Y?
Well, the common denominator would be Y.
That one doesn't change.
What do I need to do to go from here to here?
I need to multiply by Y. So I do the same thing
to the top and I get 2Y.
Now, combine those -- bottoms are the same,
and I get 2Y plus 1.
Okay, divided by again, what's the common denominator here?
Y. What do I need to do to go from this to this?
I need to multiply by Y. I do the same thing to the top,
this one doesn't change,
so now denominators stay the same and I do writers.
Now I have a fraction divided by a fraction,
which means it's going to be a fraction times the
flipped version.
Remember, you flip and multiply, all right?
Well, I have something that's being added or subtracted
so the only way I can reduce it is if I have something the same,
and I cannot do anything with it.
There's nothing I can take out, or anything like that,
so my only thing to reduce is this:
I'm left with 2Y plus 1 over 3Y minus 2.
And that's my answer.
It's just taking what we've already done --
oops, there we go, sorry, almost lost myself there,
it's taking what we've already done with adding and subtracting
and that we've already done with multiplying and dividing
and putting it all together.
All right, let's look at another one.
What if I had 4 minus 3 over X over 5 minus 1 over X,
That's an X. That is literally 4 minus 3 over X divided
by 5 minus 1 over X. Now, I can't flip anything
until I get a common denominator and put them together.
So what is the common denominator to this first bunch?
With the X. I have to multiply this by X to get the same
so I do to the bottom what I do to the top,
that one doesn't change.
So now the denominators are the same and I have 4X minus 3,
all divided by again, what's going
to be the common denominator?
Going to be X. That one doesn't change, and this one I have
to multiply by X so I multiply the top by X. So denominator has
to be the same and it stays the same,
numerators we put together.
All right?
Now I have a fraction divided by a fraction.
So I can flip and multiply.
Remember though, I have things that are being added
or subtracted, so unless I have something exactly the same I
cannot reduce them.
But I can reduce the X's, so I am left
with 4X minus 3 over 5X minus 1.
All right?
All right.
Let's go one step further.
What if I had
[ Noises ]
This, now, remember you can't have negative exponents,
so that really becomes 1
over X squared minus 2 over Y to the first.
Remember, anything with a negative exponent moves down.
Right now it's in the numerator,
so it goes down to the denominator.
This is all over Y minus 2X squared.
So what I have is 1 over X squared minus 2 over Y,
divided by Y minus 2X squared over 1.
This one is a single fraction, this one isn't.
For this one I have to change to a common denominator,
and the way you do that is what they have X's, they have Y's,
what's the highest exponent for each?
Okay? Now, to go from X squared to X squared Y
that means I multiply it by Y, I multiply the tops by Y
and I get Y. To go from Y to X squared Y means I multiplied
by X squared, I multiply the top by X squared
and I get 2X squared, all divided by okay,
so now I can put these together.
Denominators have to be the same and they stay the same, okay,
so we put those together and now it's dividing so I have
to say it -- I'm going to move up here so I have some room.
Y minus 2X squared over X squared Y times 1
over Y minus 2X squared.
Okay? Now, I put parentheses around everything, but look.
Don't you have to have parentheses around things
that are being added --
oh, that's not being added or subtracted.
Is there parentheses around things
that are being added or subtracted?
Look at it this way, Y minus 2X squared over X squared Y times 1
over Y minus 2X squared.
Are these the same?
We can reduce them.
I'm left with 1 times 1 over X squared Y. So 1
over X squared Y. And that's what 7.3 is,
it's taking complex fractions, rewriting them
as multiplication problems, you rewrite them
as division problems and then flip and multiply,
and within each piece, combine it
so you have one single fraction,
so you're putting together your addition and subtraction
and your multiplication and division.
All right.