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>> 2.2f. Reduce with variables.
When reducing with variables we need to divide out the variables
that are in common just like before
when we divided out factors or numbers that were in common.
With exponents on the variables it may help to expand them.
Expanding them simply means to write them out.
Meaning if I have x to the third,
I could write this out as x X x X x.
Let's see this in example 1.
In example 1, we have some variables with exponents
that we can start by expanding them out.
If I write this as 4 X x X x X y X z,
I can easily see each of the parts.
On the bottom I have 10 X x X y X y X y.
Now it is sometimes very helpful to use a color
other than the one that you wrote the problem in to reduce.
Otherwise, when you cross things out,
sometimes you cannot see which ones you crossed out.
I like to use a color such as red stands out clearly.
And also I will cross out the x's horizontally
so that I don't accidentally just make it look like a bigger x.
I also have a y in common and those are the only variables I have in common.
But we also have numbers.
Remember, you cannot forget the numbers when reducing with variables.
We see we have 4 and 10.
I know both of these are divisible by 2.
So we will start there.
This means 4 ÷ 2 is 2.
We also have an x and a z still there.
I sometimes like to put a line through my z's
so that I don't confuse them with 2's.
10 ÷ 2 is 5.
And we have 2 y's.
Remember, you must always put it back to the original form,
which was with exponents and make it y^2.
There are no more things in common,
therefore our answer is 2xz over 5y^2..
Some people find it easier
instead of expanding or writing out all the variables,
to instead look at the original problem.
Let's look at the x's.
Where we have x^2 on the top and x on the bottom,
they then ask themselves which side has more?
The top has more.
And then you ask yourself, how many more does it have?
It only has 1 more.
Which means the ones on the bottom are completely gone
and the one on the top has 1.
The same can be done with the y's.
We see that there are 3 y's on the bottom and 1 y on top.
Which means that the bottom has more.
How many more does it have?
It has 2 more.
Which means we have y2 on the bottom and no y's on top.
Use whichever method is best for you.
Let's look at example 2.
In example 2, we can expand it out
and we have 27 X a X a X a X b X c
all over 9 X a X a X b X b X c.
We can now start reducing variables.
We see that there's an a in common, a second a in common,
a b in common and a c in common.
We also see that both numbers are divisible by 9.
This leaves us with 27 ÷ 9 which is 3.
We also still have an a left over in the numerator.
9 ÷ 9 is 1.
And we also still have a b left over in the denominator.
A 1 is not necessary if there are other pieces in the denominator
or next to it.
Therefore we do not write the 1 and we just write this as 3a over b.
Remember, when reducing with fractions that have variables
you can expand out those variables by writing them all out
or you can use the method shown at the end of example 1
in which you reduce them in the original problem.