Chabot Mathematics
§7.5 Denom
Rationalize
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Review § 7.4
MTH 55
 Any QUESTIONS About
• §7.4 → Add, Subtract, Divide Radicals
 Any QUESTIONS About HomeWork
• §7.4 → HW-33
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Multiply Radicals
 Radical expressions often contain
factors that have more than one
term.
 Multiplying such expressions is
similar to finding products of
polynomials.
 Some products will yield like radical
terms, which we can now combine.
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Example  Multiply Radicals
 Find the Product for
3 6

5 7 7

 SOLUTION
3 6

5 7 7

 3 6  5  3 6 7 7
 3 30  21 42
Chabot College Mathematics
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Use the distributive
property.
Multiply Using Product
Rule for Radicals
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Example  Multiply Radicals

 Find the Product for 4 5  2


5 5 2 .
 SOLUTION (F.O.I.L.-like)
4
5 2

5 5 2

 4 55  5 4 5 2  5 2  5 2 2
 4  5  20 10  10  5  2
Use the product rule.
 20  20 10  10  10
Find the products.
 10  19 10
Combine like radicals.
Chabot College Mathematics
5
Use the distributive
property.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Example  Multiply Radicals

 Find the Product for
5 3

2
 SOLUTION

  5
2
5 3 
2
 2  15  3 
 5  2 15  3
 3
2
Use (a – b)2 = a2 – 2ab – b2
Simplify.
 8  2 15
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Example  Multiply Radicals


 Find the Product for 8  3 8  3

 SOLUTION


8 3 8 3

8 
2
 
 64  3
3
2
Use (a + b)(a – b) = a2 – b2.
Simplify.
 61
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Example  Multiply Radicals
 Perform
MultiTerm
Multiplication
2( y  7)
a)
b)

c)


3 x  2  3 x2  3 


m n


m n

 SOLUTION a)
a)
2( y  7)  2  y  2  7
Using the
distributive law
 y 2  14
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
a) 2( yRadicals
 7)
Example  Multiply
 Perform
MultiTerm
Multiplication
b)

c)

 SOLUTION b)
b)



3 x  2  3 x2  3 


m n
F

O

m n
I
L
3 x  2  3 x 2  3   3 x 3 x 2  33 x  2 3 x 2  6




3 3
3 2
3
x 3 x 2 x 6
3 2
3
 x3 x  2 x 6
Chabot College Mathematics
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
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
2( y  7)
a)
3 2


3
Example  Multiply
Radicals
b)  x  2   x  3 
 Perform MultiTerm
c)
Multiplication


m n


m n
 SOLUTION c)
c)

m n

m n
F

 m  (
2

O
I
L
m n  m n)
 n
mn
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
2

Radical Conjugates
 In part (c) of the last example,
notice that the inner and outer
products in F.O.I.L. are opposites,
the result, m – n, is not itself a
radical expression. Pairs of radical
terms like, m  n and m  n ,
are called conjugates.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Mult. Radicals by Special Prods
 Multiplication of expressions that
contain radicals is very similar to the
multiplication of polynomials
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12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Mult. Radicals by Special Prods
 Compare F.O.I.L. and Square of a
BiNomial-Sum
FOIL Method
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Rationalize DeNominator
 When a radical expression appears in a
denominator, it can be useful to find an
equivalent expression in which the
denominator NO LONGER contains a
RADICAL. The procedure for finding
such an expression is called
rationalizing the denominator.
 We carry this out by multiplying
by 1 in either of two ways.
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14
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Rationalize → Method-1
 One way is to multiply by 1 under the
radical to make the denominator of the
radicand a perfect power.
a)
 EXAMPLE  Rationalize Denom:
a)
5
57

7
77
5
7
3
Multiplying by 1 under the 3radical
b)
25
35
35
35



49
7
49
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Example 
5
a)
Rationalize
7
 Rationalize DeNom:
b)
3
DeNom
3
25
 SOLUTION
b) 3
3 3 3 5 Since the index is 3, we need

 3 identical factors in the denom.
25
55 5
3
Chabot College Mathematics
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15
53

3 15
3 3
5

3 15
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Rationalize → Method-2
 Another way to rationalize a DeNom is
to multiply by 1 outside the radical.
5
 EXAMPLE  Rationalize Denom:
a)
3x
5
5
3y
5
3
x
a)

b)
Multiplying
by 1


3x
3 4 xy 2
3x
3x 3x

Chabot College Mathematics
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15 x

3x

2
15 x

3x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
5
a)
3x
Example  Rationalize
3y
b)
 Rationalize DeNom:
3 4 xy 2
3y
 SOLN b)
3 4 xy 2
Need in
DeNom Radical
3
3
2 x y
Chabot College Mathematics
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3


3y

DeNom
3 2 x2 y
3 4 xy 2 3 2 x 2 y
2
3
3y 2x y
3 8 x3 y 3
3 y 3 2 x 2 y 33 2 x 2 y


2 xy
2x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Example  Rationalize DeNom
 Rationalize the denominator.
Assume variables are >0
7
3
2
16x
 SOLN  Need in DeNom Radical 43x3
3
3
3
7
7
7
4x
3



2
3
3
2
3
2
16x
4x
16x
16 x

Chabot College Mathematics
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3
3
28 x
64 x 3
3
28 x

4x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Rationalize 2-Term Rad DeNoms
 Recall that the Difference-of-2Sqs
Product results in the O & I terms in the
FOIL Multiplication Adding to Zero
 To Rationalize a DeNominator that
contains two Radical Terms requires the
use of Conjugates (which have a Diff-ofSqs form) to remove the radicals from
the Denom
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Rationalize 2-Term Rad DeNoms
 For Example to Rationalize
the Denom of
 Multiply the Numerator & Denominator
by the CONJUGATE of the Original
Denominator
5 2
45  4 2


5 2 5 2 5 2


20  4 2
5 5 2 5 2 
2
Chabot College Mathematics
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
 2
2

20  4 2 20  4 2


25  2
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Example  Rationalize DeNom
 Rationalize the denominator:
 SOLUTION
5
5
7y

.
7y
7y 7y


Chabot College Mathematics
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5


7y
7y


7y
5
.
7y
Multiplying by 1 using
the conjugate

5 7  5y
7  y2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Example  Rationalize DeNom
5 3
 Rationalize the denominator:
.
3 5
 SOLUTION
5 3
5 3
3  5 Multiplying by 1 using


3 5
3  5 3  5 the conjugate
5  3  3  5 

5


 3  5  3  5 
35 5  3 3 3 5
 3   5
2
2
5 3  5 5  3  15 5 3  5 5  3  15


35
2
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Rationalize Numerator
 To rationalize a numerator with more
than one term, use the conjugate of the
numerator
 Example  Rationalize numerator
5  3x
6
 SOLUTION
5  3x
6
5  3x 5  3x


6
5  3x
5 
2

Chabot College Mathematics
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

6 5

3x 
2
3x
25  3x

30  6 3x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
WhiteBoard Work
 Problems From §7.5 Exercise Set
• 22, 38, 64, 74, 92, 128 → Derive φ

The
Golden Ratio
φ (phi)
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
All Done for Today
L. Da Vinci
Used The
Golden Ratio
 Typo in Book for 2
1/GoldenRatio
5 1
Chabot College Mathematics
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2
5 1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
27
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
Graph y = |x|
6
 Make T-table
x
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Chabot College Mathematics
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5
y = |x |
6
5
4
3
2
1
0
1
2
3
4
5
6
y
4
3
2
1
x
0
-6
-5
-4
-3
-2
-1
0
1
2
3
-1
-2
-3
-4
-5
file =XY_Plot_0211.xls
-6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
4
5
6
5
5
y
4
4
3
3
2
2
1
1
0
-10
-8
-6
-4
-2
-2
-1
0
2
4
6
-1
0
-3
x
0
1
2
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5
-2
-1
-3
-2
M55_§JBerland_Graphs_0806.xls
-3
Chabot College Mathematics
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M55_§JBerland_Graphs_0806.xls
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BMayer@ChabotCollege.edu • MTH55_Lec-44_sec_7-5_Rationalize_Denoms.ppt
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