Rational Expressions: Basic Operations

Multiplication and Division
In algebra we multiple, divide, add, and subtract fractions the same way
as we do in arithmetic.

and then, if possible, you try to reduce
the fraction to lowest terms.

But we can make the problem easier if we notice that we can immediately
cross-cancel common factors appearing in the numerators and in the denominators,

.

We should be alert to do this when we multiply or divide algebraic fractions.
The same rules of operation apply in algebra.
When we divide fractions, we invert the divisor and multiply.
For example,

Now factor everything that we can so that we can see what common factors,
if any, can be canceled out.

The factor    appearing in the numerator and in the denominator cancel out,

.

See Examples 1 - 3, pages 34 - 36.

As in arithmetic when the denominators are not the same, we need to find the LCD,
the Lowest Common Denominator, to work out the problem.

Subtract:

First, we factor completely the denominators,

is the first denominator

is the second denominator.

Now we can write down the LCD in factored form. We will write it as a product.
If we understand the process for this relatively simple example, we will be able to
do any problem of adding or subtracting fractions. The basic idea will be the same,
no matter how many fractions are included in the problem or how complicated the
problem looks, the basic process will be the same.
Here is the process:
We must write down in an appropriate way all the factors
that appear in all the denominators:

appears as a factor in the first denominator and in the second denominator,
both times it is raised to the same power, namely, the first power, so we include it, raised
to the first, as a factor in the LCD.

appears as a factor in the first denominator only, it does not appear
as a factor in the second denominator, we include it as a factor in the LCD.

appears as factor in the second denominator only, it does not appear
as a factor in the first denominator, we include as a factor in the LCD.
Now there are no more factors that we must include in the LCD, so we are done.

There is a subtlety here that we must explain:
If a factor appears raised to the same power in any number of denominators,
it must be included as a factor raised to that same power in the LCD.
For example, if the denominators were

and
then
.

If a factor raised to different powers appears in any number of denominators,
it must be included raised to the largest power in the LCD.
If the denominators were
and
then
.

That's it! Now we will be able to write down the LCD of any number of fractions.

Let's get back to our original problem:
As in arithmetic, we must rewrite the first and the second fraction as
equivalent fractions with the LCD as their common denominator.
It is easy to do this, now that we have everything factored.

Look at the factored form of the first denominator and look at the LCD.
What factor do we see in the LCD that we don't see in the first denominator?
So we multiply it into the numerator and the denominator of the first fraction
in order to rewrite it as an equivalent fraction with the LCD as its denominator,

.

Likewise, what factor do we see in the LCD that we don't see in the second denominator?
So,

.

Now we can finish the problem:

When we combine the two fractions into one fraction, using the parentheses
to group the relevant terms in the numerator is extremely important.
Not using the parentheses may easily result in getting the wrong answer !
The next step is to drop the parentheses,

.
We must always try to reduce fractions to lowest terms.
If possible, we factor the numerator to see if we can cancel any factors common
to both the numerator and the denominator,

The factor in the top and the bottom cancel,

.                     Finished!

See Example 4, pages 37 - 38.

Complex Fractions
Another use of the LCD is to simplify complex, or compound, fractions.
A complex fraction is a fraction made up of fractions, appearing in the
numerator or in the denominator.
For example,
,     ,        are complex fractions.

In general, the easiest way to simplify a complex fraction is to multiply
the top and the bottom of the complex fraction by the LCD of its constituent fractions.
The constituent fractions are the fractions inside the big fraction.
Look at the first, algebraic, fraction above.
Explicitly writing out the denominators, the constituent fractions
on the top of the complex fraction (the big fraction) are
and .

The constituent fractions on the bottom of the complex fraction are
and .

If we were to write down all the denominators we see in the complex fraction -
without the qualification that they are to be the denominators of the constituent
fractions - we would include
.

In this method, we use the LCD of the denominators of the constituent fractions,
.

Using the distributive property,

.

Finished!

See Examples 5 - 6, pages 39 - 40.

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