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(male narrator) In this video,
we will look at simplifying expressions
that have rational exponents.
We will do so by reviewing our exponent properties.
When we have the same base multiplied together,
you will recall that we will add the exponents together.
When we have the same base divided,
we will do the opposite,
which is to subtract the exponents.
We have several power rules.
One says if the exponent is outside the parentheses,
and inside the parentheses is only multiplication or division,
we can put that exponent onto each factor:
a to the m, b to the m.
Notice, this does not work
if there was any adding or subtracting
inside the parentheses.
Similarly, with division,
we can put that exponent onto each factor:
a to the m, over b to the m.
As an exponent goes through parentheses,
we will multiply the exponents together: a to the mn.
You may also recall that anything with a 0 exponent on it
will always divide out to 1.
This does not mean an exponent of 1, a to the 1st.
It means simply the number 1.
Negative exponents mean the base has to move,
either to the denominator in this first example
or back up to the numerator in the second example.
If the negative exponent is on the fraction,
we can simply flip the fraction to its reciprocal
and then deal with that exponent.
When we're simplifying
and using several properties in the same problem,
we will remember the order of operations...
to help guide us
through simplifying the expression.
These problems are very similar
to what we did back when we did exponent properties.
The only difference now are the exponents are fractions.
In this problem, we will want to simplify the numerator first,
because of the invisible parentheses
around the numerator and the denominator.
When we try and combine the x's together
into a single x in the numerator,
what we will want to do is add those exponents.
Adding exponents may be difficult to do mentally,
so let's put some scratch work off to the side.
We're adding 4/3 plus 5/4.
To add, we need a common denominator of 12.
Multiplying the first fraction by 4 and the second by 3,
we get 16/12, plus 15/12, which is 31/12.
This is the new exponent on x.
Similarly with the y's-- y to the 2/7 and y to the 3/7--
we want to add those exponents.
Fortunately, we already have a common denominator:
2/7 plus 3/7 is 5/7.
There is nothing to simplify in the denominator.
We have x to the 1/2, y to the 2/7.
Now, to simplify the division,
we will want to subtract the exponents on the x
to get the new exponent on x: 31 over 12, minus 1/2,
will have a common denominator of 12.
We multiply it by 6 over 6 to get 31/12, minus 6/12,
and 31 minus 6 is 25/12.
This is my new exponent on the x.
Similarly, on the y's, we can subtract those exponents...
whoops, I think I wrote 5/5.
I meant to write 5/7 on there: 2/7 minus 3/7.
Now, we are ready to subtract the exponents 5/7 minus 2/7,
which we could probably do mentally
by subtracting the numerators: 5 minus 2 is 3/7.
We are done simplifying when each variable appears once,
and there are no negative exponents in our solution.
In our next video, Part 2, we'll simplify another example,
this time with negative exponents.