Chabot Mathematics
§7.3 Factor
Radicals
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Review § 7.3
MTH 55
 Any QUESTIONS About
• §7.3 → Multiply Radicals
 Any QUESTIONS About HomeWork
• §7.3 → HW-32
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Product Rule for Radicals
nnb ,
 For any real numbers nnaa and
and
and b ,
n a  n b  n a  b.
 That is, The product of two nth roots
is the nth root of the product of the
two radicands.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Simplifying by Factoring
 The number p is a perfect square if
there exists a rational number q for
which q2 = p. We say that p is a
perfect nth power if qn = p for some
rational number q.
 The product rule allows us to
simplify n ab whenever ab contains
a factor that is a perfect nth power
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Simplify by Product Rule
 Use The Product Rule in
REVERSE to Facilitate the
Simplification process
n ab  n a  n b
• Note that nnaa and
and nnbb must both be
and
real numbers
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Simplify a Radical Expression
with Index n by Factoring
1. Express the radicand as a product
in which one factor is the largest
perfect nth power possible.
2. Take the nth root of each factor
3. Simplification is complete when
no radicand has a factor that is a
perfect nth power.
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Example  Simplify by Factoring
 Simplify by factoring (assume x > 0)
a)
b)
 SOLUTION → Match INDICES
a)
Chabot College Mathematics
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b)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Example  Simplify by Factoring
 Simplify by factoring (assume x > 0)
a)
b)
 SOLN a)
Note
That the
INDEX
is 3
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Example  Simplify by Factoring
 Simplify by factoring (assume x > 0)
a)
b)
 SOLN b) Note INDEX of 5
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Example  Simplify by Factoring
a) (assume
10 6 x, y > 0)
 Simplify by factoring
a)a) 10 6
b) 3 9 x3 y 2 3 9 x 4 y 7
b)
3 23 4 7
3
9 xa)y Note
9 x INDEX
y
 b)
SOLN
of 2
a)
10 6  60
 4 15
 2 15
Chabot College Mathematics
10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Example  Simplify by Factoring
a) (assume
10 6 x, y > 0)
 Simplify by factoring
a)a) 10 6
b) 3 9 x3 y 2 3 9 x 4 y 7
b)
3 23 4 7
3
9 x 3y 3 92x 3 y 4 7 3
7 9
 b)
SOLN
b) 9 x y 9 x y  81x y
b)
Note
INDEX
of 3
 3 27  3  x6  x  y 9
3 6 3 9 3
3
 27  x  y  3 x
 3x 2 y3 3 3x
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Example  Simplify
 Simplify by factoring (assume w, z > 0)
5
 2wz
4 6 5
8w z
7
 4w z
5
6 9

 SOLN: First perform Distribution
• Note that all INDICES are common at 5
5
 2wz
4 6 5
8w z
 8w z
5
4 65
Chabot College Mathematics
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7
 4w z
5
6 9
2wz  8w z
7
5

4 65
6 9
4w z
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Example  Simplify
 SOLN: Use Radical Product Rule
5
4 65
8w z

5
2wz  8w z
7
5
4 65
6 9
4w z
8w z  2wz   8w z  4w z 
4 6
7
5
4 6
6 9
 SOLN: Use Commutative Property of
Multiplication
8w z  2wz   8w z  4w z 
 8  2w  wz  z   8  4w  w z
4 6
5
5
Chabot College Mathematics
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7
4
5
6
7
4 6
6 9
5
4
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
6
z
9

Example  Simplify
 SOLN: Exponent Product Rule
5
8  2w
4

 8  4w
w z z 
6
7
5
4

w z z
6
6
9

 5 16w41 z 67  5 32w4 6 z 69
 5 2 4 w5 z13  5 25 w10 z15
 SOLN: Next use Exponent POWER rule
to expose as many bases as possible
to the Power of 5; the Radical Index
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
Example  Simplify
 SOLN: Power-to-Power Exponent Rule
5
2 wz  2 w z
4
5 13
5
10 15
 z
 16w z
5
5
5
2 5
3

5
2 w
5
 z 
2 5
3 5
 SOLN: Next Radical Product Rule
5
 z
16w z
5
 16 z 
5
3
2 5
5
Chabot College Mathematics
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3
w
5


5
5
2 w  z 
2 5
5
z 
2 5

5
3 5
2
5

5
w   z 
2 5
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
3 5
Example  Simplify
 SOLN: Perform 5th Root Operations
16 z 
5
3
5
w
5

5
z 
2 5

5
2
5

5
w   z 
2 5
5
3 5
 5 16 z 3  w  z 2  2  w2  z 3  2w2 z 3  wz 2 5 16 z 4
 SOLN: Finally Factor GCF = wz2
2w z  wz
2 3

25
16 z
3
 wz 2wz  16 z
2
Chabot College Mathematics
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5
3

ANS
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
WhiteBoard Work
 Problems From
§7.3 Exercise Set
• 76, 80, 82, 92, 98

Adult
Cardiac
Index =
2.8-3.4
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
All Done for Today
Exponent
Rules are
NOT
Algebraic
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.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
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.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
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file =XY_Plot_0211.xls
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.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
3
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5
-2
-1
-3
-2
M55_§JBerland_Graphs_0806.xls
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Chabot College Mathematics
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M55_§JBerland_Graphs_0806.xls
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-42_sec_7-3b_Factor_Radicals.ppt
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