Multiplying Rational Numbers Examples
1. To multiply rational numbers (fractions,) you multiply the numerators and
multiply the denominators.
3 4
3 • 4 12
Î
• Î
5 7
5•7
35
Multiplying
Fractions
This method can be generalized as follows:
a
c
and , where b ≠ 0 and d ≠ 0,
b
d
a c ac
• =
.
b d bd
For all rational numbers
This same method can be used to multiply rational expressions.
2. Example: Find
5 b
• and simplify. State any excluded values.
a 7
Multiply the numerators.
Multiply the denominators.
5 b 5 • b 5b
• =
=
a 7 a • 7 7a
Since 7a cannot equal
zero, a ≠ 0.
3. Example: Find
2a 2 d 9b 2c
•
and simplify. State any excluded values.
3bc 16ad 2
2a 2 d 9b 2c
18a 2b 2cd 3ab
•
=
=
3bc 16ad 2 48abcd 2
8d
The GCF is 6abcd.
Change the fraction to simplest form.
Since bc and ad2 cannot equal 0, a ≠ 0, b ≠ 0, c ≠ 0, and d ≠ 0.
"Multiplying Rational Numbers"
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Rev.7.8.03
4. Thought ProvokerÎFind a value for “b” so that
a+b a
= .
b+c c
b=0
5. Thought Provoker ÎFind values for “b” so that
ab a
= .
bc c
b = any real
number except 0.
6. Review students of the “Cross-Product” rule:
This same method can be
used with rational
expressions.
1 4
3 16 3 16 4
•
= •
=
4 21 4 21 7
1 7
12 x 2
x+5
7. Example: Find
• 2
and simplify.
3x
x + 7 x + 10
x ≠ 0, -5, or -2
x+5
12 x 2
x + 5 2• 2•3• x • x
•
• 2
=
3x
x + 7 x + 10 3 • x ( x + 5)( x + 2)
Simplify – stress that each
polynomial must be written in
factored form and then simplified.
x + 5 2• 2•3• x • x
•
3 • x ( x + 5)( x + 2)
4x
x+2
A common error that students make
in simplifying is show here.
"Multiplying Rational Numbers"
2 1
4x
2
= =1
x+2 2
1 1
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Rev.7.8.03
8. Example: Find
a −5
4a + 8
•
and simplify.
2
a − 25 5a + 10
a −5
4a + 8
4( a + 2)
1(a − 5)
•
=
•
2
a − 25 5a + 10 (a − 5)(a + 5) 5(a + 2)
1
1
4( a + 2)
1(a − 5)
•
(a − 5)(a + 5) 5(a + 2)
1
1
4
5(a + 5)
a ≠ 5, -5, or -2
Simplify
4
5a + 25
x 2 − x − 6 x 2 + 7 x + 12
9. Example: Find
and simplify.
• 2
9 − x2
x + 4x + 4
x 2 − x − 6 x 2 + 7 x + 12 ( x − 3)( x + 2) ( x + 3)( x + 4)
( x − 3)( x + 2) ( x + 3)( x + 4)
•
=
•
=
• 2
2
9− x
x + 4 x + 4 (3 − x)(3 + x) ( x + 2)( x + 2) − 1( x − 3)( x + 3) ( x + 2)( x + 2)
1
1
1
( x − 3)( x + 2) ( x + 3)( x + 4)
•
− 1( x − 3)( x + 3) ( x + 2)( x + 2)
x ≠ 3, -3, or -2
1
1
1
x+4
Notice that
3 – x = -1(x – 3)
− 1( x + 2)
−
x+4
x+2
Simplify
Caution students to write the negative sign before the
fraction bar and not with the numerator. For example,
x+4
−x+4
−
is not the same as
.
x+2
x+2
"Multiplying Rational Numbers"
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Rev.7.8.03
Name:___________________
Date:___________
Class:___________________
Multiplying Rational Numbers Worksheet
Find each product and simplify (Exclude values if necessary)
1.
2 3
•
9 5
9.
2.
ab c
•
ac d
3.
y − 3 14
•
7
y −3
4.
− ( 2a + 7c )
36
•
6
− 7c − 2a
5.
3
( x − y)2
•
6
x− y
6.
9
m2 − 9
•
m−3
12
7.
x 2 − 16 x + 4
•
9
x−4
8.
b 2 + 20b + 99
b+7
• 2
b+9
b + 12b + 11
3a − 6
a −9
2
•
b 2 + 19b + 84
b2 − 9
10.
• 2
b−3
b + 15b + 36
a+3
a 2 − 2a
"Multiplying Rational Numbers"
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Rev.7.8.03
Multiplying Rational Numbers Worksheet Key
Find each product and simplify.
1.
2 3
•
9 5
ab c
•
2.
ac d
1
2
3
2
• Î
3•3 5
15
1
1
1
ab c
b
• Î
ac d
d
1 1
a ≠ 0, c ≠ 0, and d ≠ 0
y ≠ 3
3.
y − 3 14
•
7
y −3
1
1
1( y − 3)
7•2
•
Î2
7
( y − 3) • 1
1
1
c ≠ −
4.
− ( 2a + 7c )
36
•
6
− 7c − 2a
5.
3
( x − y)2
•
x− y
6
"Multiplying Rational Numbers"
2a
7
1
1
1
6•6
− 1(2a + 7c)
•
Î6
1• 6
− 1(7c + 2a )
1
1
1
1
1
1
3
( x − y )( x − y )
x− y
•
Î
1 • ( x − y)
3• 2
2
1
1
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x ≠ y
Rev.7.8.03
m ≠ 3
6.
m −9
9
•
m−3
12
7.
x 2 − 16 x + 4
•
9
x−4
2
8.
9.
3a − 6
a2 − 9
•
1
1
3•3
(m − 3)(m + 3) 3m + 9
•
Î
1 • (m − 3)
3• 4
4
1
1
a+3
a 2 − 2a
1
x 2 + 8 x + 16
( x + 4)( x − 4)
x+4
•
Î
9
9
1( x − 4)
1
1
1
3(a − 2)
1 • (a + 3)
3
•
Î 2
(a + 3)(a − 3) a(a − 2)
a − 3a
1
1
b 2 + 20b + 99
b+7
• 2
b+9
b + 12b + 11
b ≠ -9, -1, or -11
10.
b 2 + 19b + 84
b2 − 9
• 2
b−3
b + 15b + 36
b ≠ 3, -3, or -12
"Multiplying Rational Numbers"
x ≠ 4
a ≠ 3, -3, 0, or 2
1
1
(b + 9)(b + 11)
1 • (b + 7)
b+7
•
Î
1 • (b + 9)
(b + 11)(b + 1)
b +1
1
1
1
1
1
(b + 7)(b + 12) (b − 3)(b + 3)
•
Îb + 7
1 • (b − 3)
(b + 3)(b + 12)
1
1
1
©2001-2003www.beaconlearningcenter.com
Rev.7.8.03
Student Name: __________________
Date: ______________
Multiplying Rational Numbers Checklist
1. On question 1, did the student find the product and simplify correctly?
a. Yes (10 points)
b. Found correct product but did not simplify correctly (5 points)
2. On question 2, did the student find the product and simplify correctly?
a. Yes (10 points)
b. Found correct product but did not simplify correctly (5 points)
3. On question 3, did the student find the product and simplify correctly?
a. Yes (10 points)
b. Found correct product but did not simplify correctly (5 points)
4. On question 4, did the student find the product and simplify correctly?
a. Yes (10 points)
b. Found correct product but did not simplify correctly (5 points)
5. On question 5, did the student find the product and simplify correctly?
a. Yes (10 points)
b. Found correct product but did not simplify correctly (5 points)
6. On question 6, did the student find the product and simplify correctly?
a. Yes (10 points)
b. Found correct product but did not simplify correctly (5 points)
7. On question 7, did the student find the product and simplify correctly?
a. Yes (10 points)
b. Found correct product but did not simplify correctly (5 points)
8. On question 8, did the student find the product and simplify correctly?
a. Yes (10 points)
b. Found correct product but did not simplify correctly (5 points)
"Multiplying Rational Numbers"
©2001-2003www.beaconlearningcenter.com
Rev.7.8.03
9. On question 9, did the student find the product and simplify correctly?
a. Yes (10 points)
b. Found correct product but did not simplify correctly (5 points)
10. On question 10, did the student find the product and simplify correctly?
a. Yes (10 points)
b. Found correct product but did not simplify correctly (5 points)
Total Number of Points _________
A
90 points and above
B
80 points and above
C
70 points and above
D
60 points and above
F
59 points and below
"Multiplying Rational Numbers"
Any score below C
needs
remediation!
©2001-2003www.beaconlearningcenter.com
Rev.7.8.03