(male narrator) In this video,
we will simplify compound fractions
that involve negative exponents.
You may recall
that a negative exponent will create a fraction.
Be careful that the negative exponent
only affects or moves what it's physically attached to.
This means, 5x to the -3 would be 5 over x cubed.
We also must be careful
that if there is any adding or subtracting,
we can't just move it.
Instead, we must make little fractions...
or compound fractions.
Let's take a look at some examples
where we have negative exponents creating a compound fraction.
In this problem, each negative exponent
will create a little fraction in its place.
This gives us 1 plus 10, over x to the first,
plus 25, over x squared.
Notice only the variable moves,
because that's the only part with the negative exponent.
In the denominator, we have 1 minus 25, over x squared.
We can now simplify this expression
by identifying the least common denominator
using the highest exponent.
We will multiply each term by x squared.
This includes the numbers that are not fractions.
When we reduce 1x with the x squared,
we're left with a single x.
The x squareds will divide out completely,
and we end up with x squared, plus 10x, plus 25,
over x squared, minus 25.
Be careful as we try and reduce that we cannot reduce
unless the expression is first factored.
The numerator factors
to x plus 5 squared,
and the denominator factors
to x plus 5, times x minus 5.
To review factoring, watch some of the previous videos
on factoring trinomials and special products.
We can see that we can, in fact,
reduce out the common factor
of 1x plus 5
from the denominator
and 1 from the numerator,
giving us our final answer:
x plus 5, over x minus 5.
Let's take a look at another example,
which uses negative exponents in order to simplify.
Again, in this problem,
we see several parts with negative exponents.
Each term will create its own little fraction.
We have 8, over b cubed; plus 27, over a cubed;
all over 4; over ab cubed; minus 6;
over a squared, b squared; plus 9; over a cubed b.
When we identify the least common denominator,
we always use our highest exponents,
which will give us a cubed, b cubed.
We will multiply each term by a cubed, b cubed,
so that when we simplify,
we get rid of all the small fractions.
Each term gets multiplied by a cubed, b cubed,
and we start reducing.
The b cubes divide out; the a cubes divide out;
and the denominator b cubed divides out;
and one a divides out, leaving two behind;
a squared divides out, leaving one behind;
b squared divides out, leaving one behind;
the a cubes divide out;
and the b divides out, leaving two behind.
We now have 8a cubed, plus 27b cubed,
over 4a squared, minus 6ab,
plus 9b squared.
You may recognize that numerator factors as a sum of cubes:
2a plus 3b, times 4a squared,
minus 6ab, plus 9b squared,
over the denominator 4a squared,
minus 6ab, plus 9b squared.
Sure enough, that trinomial divides out,
and our final answer is 2a plus 3b.
