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Dividing Monomials
November 4, 2011
Dividing Monomials
Objective To simplify quotients of
monomials and to find the
greatest common factor (GCF) of
several monomials.
Dividing Monomials
There are three basic rules used to simplify
fractions whose denominators and
numerators are monomials. The property of
quotients allows you to express a fraction as
a product.
Property of Quotients
If a, b, c, and d are real numbers with 𝑏 ≠ 0
and 𝑑 ≠ 0, then
𝑎𝑐 𝑎 𝑐
= ∙
𝑏𝑑 𝑏 𝑑
Example 1
15 3 ∙ 5 3 5
5 5
=
= ∙ =1∙ =
21 3 ∙ 7 3 7
7 7
Simplifying Fractions
You obtain the following rule for simplifying
fractions if you let 𝑎 = 𝑏 in
the property
of quotients.
If b, c, and d are real numbers with
𝑏 0 and 𝑑 0, then
𝑏𝑐 𝑐
=
𝑏𝑑 𝑑
Simplifying Fractions
The following rule allows you to divide the
numerator and the denominator of a
fraction
by the same nonzero number. In
the examples of this lesson, assume that no
denominator equals zero.
Example 2
Simplify
35
42
solution
35 5 ∙ 7 5 7 5
=
= ∙ =
42 6 ∙ 7 6 7 6
Example 3
Simplify
𝑐7
𝑐5
Simplify
𝑐3
𝑐8
solution
solution
𝑐7 𝑐5 ∙ 𝑐2
2
=
𝑐
=
𝑐5
𝑐5
1
𝑐3
𝑐3
=
=
𝑐8 𝑐3 ∙ 𝑐5 𝑐5
Rule of Exponents for Division
Remember: when you multiply powers, you
add the exponents.
The results of Example 3 show that when
you divide powers with the same base, you
can subtract the smaller exponent from the
greater, if they are different.
Rule of Exponents for Division
If a is a nonzero real number and m and n
are positive integers, then
If 𝑚 > 𝑛:
If 𝑚 < 𝑛:
If 𝑚 = 𝑛:
𝑎𝑚
𝑚−𝑛
=
𝑎
𝑎𝑛
𝑎𝑚
1
= 𝑛−𝑚
𝑛
𝑎
𝑎
𝑎𝑚
=1
𝑛
𝑎
Example 4
Simplify
𝑥9
𝑥5
Simplify
𝑥2
𝑥7
solution
solution
𝑥9
9−5
4
=
𝑥
=
𝑥
𝑥5
1
𝑥2
1
= 7−2 = 5
7
𝑥
𝑥
𝑥
The Greatest Common Factor
The greatest common factor (GCF) of two
or more monomials is the common factor
with the greatest coefficient and the
greatest degree in each variable.
Example 5
Find the GCF of 72x3yz3 and 120x2z5
solution
Step 1
Find the GCF of the numerical
coefficients.
72 = 23 ∙ 32
and
120 = 23 ∙ 3 ∙ 5
the GCF of 72 and 120 is 233=83=24.
Example 5
Find the GCF of 72x3yz3 and 120x2z5
Solution
Step 2
Find the smaller power of each
variable that is a factor of both
monomials.
The smaller power of x is x2
y is not a common factor.
The smaller power of z is z3
Example 5
Find the GCF of 72x3yz3 and 120x2z5
solution
Step 3
Find the product of the GCF of the numerical
coefficients and the smaller power of each
variable that is a factor of both monomials.
24 · x2 · z3
 the GCF of 72x3yz3 and 120x2z5 is 24 · x2 · z3 .
Simplified Quotient of Monomials
A quotient of monomials is said to be
simplified when each base appears only
once, when there are no powers of powers,
and when the numerator and
denominator
have no common factors other than 1.
Example 6
Simplify
35𝑥 3 𝑦𝑧 6
56𝑥 5 𝑦𝑧
Solution 1
Use the property of
quotients and the rule of
exponents for division.
35𝑥 3 𝑦𝑧 6 35 𝑥 3 𝑦 𝑧 6
=
∙ 5∙ ∙
5
56𝑥 𝑦𝑧
56 𝑥 𝑦 𝑧
5
5 1
5𝑧
= ∙ 2 ∙ 1 ∙ 𝑧5 =
8 𝑥
8𝑥 2
Example 6
Solution 2
Find the GCF of the numerator and
denominator and use the rule for
simplifying fractions.
35𝑥 3 𝑦𝑧 6 7𝑥 3 𝑦𝑧 ∙ 5𝑧 5 5𝑧 5
= 3
= 2
2
5
8𝑥
56𝑥 𝑦𝑧 7𝑥 𝑦𝑧 ∙ 8𝑥
Class work
Homework
P 191 Oral Exercises: P 192: 1-57 odd
1-36
P 193: Mixed Review