Lesson 22
A STORY OF FUNCTIONS
M1
ALGEBRA II
Lesson 22: Equivalent Rational Expressions
Classwork
Opening Exercise
1
On your own or with a partner, write two fractions that are equivalent to , and use the slips of paper to create visual
3
models to justify your response.
Example 1
Consider the following rational expression:
of 𝑎𝑎 is the expression undefined?
Lesson 22:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
2(𝑎𝑎−1)−2
6(𝑎𝑎−1)−3𝑎𝑎
. Turn to your neighbor and discuss the following: For what values
Equivalent Rational Expressions
S.106
Lesson 22
A STORY OF FUNCTIONS
M1
ALGEBRA II
Exercise 1
Reduce the following rational expressions to lowest terms, and identify the values of the variable(s) that must be
excluded to prevent division by zero.
a.
b.
c.
d.
2(𝑥𝑥+1)+2
(2𝑥𝑥+3)(𝑥𝑥+1)−1
𝑥𝑥 2 −𝑥𝑥−6
5𝑥𝑥 2 +10𝑥𝑥
3−𝑥𝑥
𝑥𝑥 2 −9
3𝑥𝑥−3𝑦𝑦
𝑦𝑦 2 −2𝑥𝑥𝑥𝑥+𝑥𝑥 2
Lesson 22:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Equivalent Rational Expressions
S.107
Lesson 22
A STORY OF FUNCTIONS
M1
ALGEBRA II
Lesson Summary
If 𝑎𝑎, 𝑏𝑏, and 𝑛𝑛 are integers with 𝑛𝑛 ≠ 0 and 𝑏𝑏 ≠ 0, then
𝑛𝑛𝑛𝑛 𝑎𝑎
= .
𝑛𝑛𝑛𝑛 𝑏𝑏
The rule for rational expressions is the same as the rule for integers but requires the domain of the
rational expression to be restricted (i.e., no value of the variable that can make the denominator of the
original rational expression zero is allowed).


Problem Set
1.
Find an equivalent rational expression in lowest terms, and identify the value(s) of the variable that must be
excluded to prevent division by zero.
a.
b.
c.
d.
e.
f.
g.
h.
2.
𝑦𝑦−𝑥𝑥
i.
20𝑛𝑛
𝑥𝑥 3 𝑦𝑦
𝑥𝑥−𝑦𝑦
j.
𝑦𝑦 4 𝑥𝑥
2𝑛𝑛−8𝑛𝑛2
k.
4𝑛𝑛
𝑑𝑑𝑑𝑑+𝑑𝑑𝑑𝑑
𝑥𝑥 2 −9𝑏𝑏 2
m.
𝑥𝑥 2 −2𝑥𝑥𝑥𝑥−3𝑏𝑏 2
3𝑛𝑛2 −5𝑛𝑛−2
n.
2𝑛𝑛−4
3𝑥𝑥−2𝑦𝑦
o.
9𝑥𝑥 2 −4𝑦𝑦 2
4𝑎𝑎2 −12𝑎𝑎𝑎𝑎
p.
𝑎𝑎2 −6𝑎𝑎𝑎𝑎 +9𝑏𝑏 2
𝑏𝑏+𝑎𝑎
4𝑥𝑥−2𝑦𝑦
3𝑦𝑦−6𝑥𝑥
9−𝑥𝑥 2
l.
𝑑𝑑𝑑𝑑
𝑎𝑎2 −𝑏𝑏 2
(𝑥𝑥−3)3
𝑥𝑥 2 −5𝑥𝑥+6
8−2𝑥𝑥−𝑥𝑥 2
𝑎𝑎−𝑏𝑏
𝑥𝑥𝑥𝑥−𝑥𝑥𝑥𝑥−𝑎𝑎+𝑏𝑏
(𝑥𝑥+𝑦𝑦)2 −9𝑎𝑎2
2𝑥𝑥+2𝑦𝑦−6𝑎𝑎
8𝑥𝑥 3 −𝑦𝑦 3
4𝑥𝑥 2 −𝑦𝑦 2
Write a rational expression with denominator 6𝑏𝑏 that is equivalent to
a.
3.
16𝑛𝑛
𝑎𝑎
𝑏𝑏
.
𝑎𝑎
b.
one-half of .
c.
1
3
.
𝑏𝑏
Remember that algebra is just another way to perform arithmetic, but with variables replacing numbers.
2
�𝑥𝑥 2 𝑦𝑦� (𝑥𝑥𝑥𝑥)3 𝑧𝑧 2
a.
Simplify the following rational expression:
b.
Simplify the following rational expression without using a calculator:
c.
How are the calculations in parts (a) and (b) similar? How are they different? Which expression was easier to
simplify?
Lesson 22:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
(𝑥𝑥𝑦𝑦 2 )2 𝑦𝑦𝑦𝑦
Equivalent Rational Expressions
.
122 ∙63 ∙52
182 ∙15
.
S.108