We're asked to subtract and state the difference
in simplest form.
So we're subtracting one rational expression from another.
And just like the case when you are subtracting fractions,
or adding fractions for that matter,
you have to make sure that you have a common denomintator.
And over here it looks like we have kind of the same components,
but they're not quite the same thing.
So we need to find the least common multiple of 7m^2n and 3mn^4,
and then that we can use as a common denominator.
Let's see if we can do that.
So, let me find a common denominator:
we have a negative sign out here and a common denominator over here.
So a common denominator is going to have ...
let's just look at each of the numbers.
We have a 7 here and a 3 over here.
So the common denominator has to be divisible by both 7 and 3.
Or another way to think about it,
the least common multiple of 7 and 3 is 21.
So we're going to have to have 21 in our common denominator.
And I'm getting that by encorporating the 7 and the 3.
That's what gives me the 21.
Now let's look at the m's.
This denominator has 2 "m"s if you were to factor it.
It's m times m.
This only has one m.
So as long as we're divisible by m^2
then we're also going to be divisible by m
So we just have to put an m^2 over here
and an m^2 over here.
Notice this is ... right now this is both divisible by
7, 3, m and m^2.
So we're cool through the "m"s.
Now let's look at the "n" terms.
It has to be divisible by n,
and it also has to be divisible by n^4.
Well, if we're divisible by n^4,
we're also divisible by n, since this is one of the factors of n^4
This to the fourth power is n to the fourth.
Anyway, we have to be divisible by n^4
If we're divisible by n^4 we are divisible by n
So that's our common denominator.
We're divisible by 7m^2n, definitely divisible by 3mn^4.
So that's our common denominator.
So to go from 7m^2n, to this right over here,
we have to multiply it by 3 times
let's see, the m^2, we didn't have to multiply it by anything
to get from n to n^4, we have to multiply it by n^3
So we have to multiply this by 3n^3
So we multiply the denominator by 3n^3 to get this over here
we also have to multiply the numerator by 3n^3
3n^3 times 6m is 18m,
one right over there
and then we have an n^3
now let's do it with the second term right over here
or the thing we are subtracting from this first rational expression
so to get from 3m,
so let's see, to get from that to that
what did we have to do?
to get from a 3 to a 21
you have to multiply it by a 7
you have to multiply it by 7
and to get from an m to an m^2 you have to multiply it by m
And to get from n^4 to n^4, you didn't have to multiply by anything
So to get from this expression in the denominator
to this expression in the denominator
you have to multiply it by 7m.
So if you multiply the denominator by 7m
you also have to ultiply the numerator by 7m
And so you have 5 time 7 is 35
and you have 1m, right over here,
and then we have n^3
and now we are ready to subtract
since we have the same denominator
Let me write the denominator here
So the denominator is 21m^2n^4
And in our numerator we have
18mn^3 minus 35mn^3
Now this is interesting,
because we have 18 of something
of mn^3
and we're going to subtract 35 of that same something
mn^3
we have the same power on m,
same power on n
So this is 18 of something -35 of that something
which is going to be equal to what?
18 minus 35,
this is just practicing subtracting negative numbers
is -17
Let me do this in a new color
So this is going to be -17
-17mn^3
right, 18-35: 35-18 would be 17
So if you swap them around you get -17
yup that works
and then all of that over 21m^2n^4.
Now let's see if we can simplify this further
both the numerator ...
17 is prime, so you can't simplify that with 21
both the numerator and denominator are divisible by m
So let's divide them both by m
this becomes a 1, this becomes just an m
both the numerator and denominator are divisible by n^3
so you divide the numerator by n^3, that becomes 1
you divide the denominator by n^3, this becomes an n
We are left with, in simplest form,
-17 over, I'll do it in the same colors, 21mn.
And we are done.