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(male narrator) In this video,
we will look at how we can multiply and divide
rational expressions made up of monomials.
When we're multiplying and dividing
with rational expressions of monomials,
we can use exponent properties...
because there is no adding or subtracting in the problem.
If you remember, the product rule says
we can add the exponents on the base.
The quotient rule says
we will subtract the exponents on the base.
Let's take a look at some problems where we do just that.
In this problem, we can multiply across the numerator:
6 times 10 to get 60;
and add the exponents on the x:
4 plus 2 is x to the 6th.
We also have a y to the 5th.
In the denominator,
we can multiply 5 times 3 to get 15;
and add the exponents on the x's:
3 plus 2 gives us x to the 5th.
We also have a y to the 7th.
Now, we can look at reducing this fraction
by dividing out common factors.
Sixty and 15 are both divisible by 15;
15 goes into 60, 4 times; and to 15, once.
We can now reduce the variables by subtracting the exponents.
On the x's, 6 minus 5 gives us x to the 1st.
Because it's positive, we put it in the numerator,
and we do not need to write the 1.
On the y's, subtracting 5 minus 7 will give us -2.
This tells us that the y needs to go in the bottom--
or the denominator--
as a result of the negative exponent,
which will now become positive in the denominator.
We also don't need the 1 in front of the y,
because the 1 is always implied.
This expression is now completely simplified.
Let's take a look at another example
where we use our exponent properties
to help us simplify an expression
that has no adding or subtracting.
In this problem, we are dividing.
This means we will have to multiply by the reciprocal.
We now have 4a to the 5th b; over 9a to the 4th;
times the reciprocal, which is 12b squared;
over 6a; b to the 4th.
Now this problem can solve
exactly like the previous problem.
Multiplying across will give us 48; a to the 5th;
on the b's, we add the exponents of 1 and 2 to get b cubed.
In the denominator, we can multiply straight across
to get 9 times 6, or 54;
a to the 5th, which we get from adding 4 plus 1;
b to the 4th.
We're now ready to reduce this expression
by dividing out common factors:
48 and 54 both have a common factor of 6;
48 divided by 6 is 8; and 54 divided by 6 is 9.
Notice the a to the 5ths are identical
in the numerator and denominator and can divide out completely.
This is why a to the 0 is 1,
because they divide out completely:
5 minus 5 is a to the 0, or 1.
On the b's, we subtract the exponents 3 minus 4 to get -1.
Because it is negative,
that tells us the b to the 1st must move to the denominator,
and we do not need to write the 1st power.
Our final answer is 8 over 9b.
Using exponent properties
can help us simplify rational expressions
when there is no adding or subtracting
anywhere in the problem.