Section 7.2
Multiplying and Dividing
Rational Expressions
Recall that one way to multiply two fractions is to
multiply numerators and multiply denominators
and then reduce to lowest terms, if possible:
Multiply numerators
Multiply denominators
Reduce to lowest terms
Recall also that the same result can be achieved by
factoring numerators and denominators first and
then dividing out the factors they have in
common:
Factor
Multiply numerators
Multiply denominators
Divide out common factors
Multiply
We begin by factoring numerators and
denominators as much as possible.
Then we write as one numerator over one
denominator.
The last step consists of dividing out all factors common to
the numerator and denominator:
Factor completely
Write as one
numerator over one
denominator
Divide out common factors
We have defined division as the equivalent of
multiplication by the reciprocal.
This is how it looks with fractions:
Division as multiplication by the reciprocal
Factor and divide out common factors
The same idea holds for division with rational
expressions.
The rational expression that follows the division
symbol is called the divisor ; to divide, we multiply
by the reciprocal of the divisor.
Divide
Steps for Solving:
1. Take the reciprocal of the divisor
2. Rewrite the problem again in terms of
multiplication.
3. Factor numerator
4. Factor denominator
5. Write as one rational expression
6. Cancel factors common to the numerator and
denominator
The complete solution looks like this:
Multiply by the reciprocal
of the divisor
Factor
Multiply
Divide out common factors
As you can see, factoring is the single most
important tool we use in working with rational
expressions.
Most of the work we have done or will do with
rational expressions is accomplished most easily if
the rational expressions are in factored form.
Unit Analysis
Unit analysis is a method of converting between
units of measure by multiplying by the number 1.
A plane has reached its cruising altitude of 35,000
feet. How many miles is the plane above the
ground? (1 mile is 5,280 feet)
By using unit analysis:
We treat the units common to the numerator and
denominator in the same way we treat factors
common to the numerator and denominator; common
units can be divided out, just as common factors are.
In the previous expression, we have feet common to
the numerator and denominator.
Dividing them out leaves us with miles only.
Here is the complete solution:
The expression
is called a conversion factor.
It is simply the number 1 written in a convenient form.
Because it is the number 1, we can multiply any other
number by it and always be sure we have not changed
that number.
The key to unit analysis is choosing the right
conversion factors.
The Mall of America in the Twin Cities covers 78
acres of land. If 1 square mile = 640 acres, how
many square miles does the Mall of America
cover?
We are starting with acres and want to end up with
square miles.
We need to multiply by a conversion factor that will allow
acres to divide out and leave us with square miles:

39
square miles
320
(This is 0.12square miles rounded to the nearest hundredth)
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Section 7.2
Page 507 – 514
# 1, 5, 9, 13, 17, 19, 25, 29, 35, 39