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Chabot Mathematics
§7.3 Radical
Products
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
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Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt
Review § 7.2
MTH 55
 Any QUESTIONS About
• §7.2 → Rational Exponents
 Any QUESTIONS About HomeWork
• §7.2 → HW-25
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Bruce Mayer, PE
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Multiplying Radical Expressions
 Note That:
4  9  2  3  6.
4  9  36  6.
 This example suggests the
following.
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Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.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.
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Bruce Mayer, PE
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Derive Product Rule for Rads
 Rational exponents can be used
to derive the Product Rule for
Radicals:
n a  n b  a1/ n  b1/ n  a  b 1/ n  n a  b.
 
n a  n b  n a  b.
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Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt
Example  Product Rule
 Multiply
a)
5 6
b) 3 7  3 9
c) 4
x 45

3 z
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 SOLUTION
a)
5  6  5  6  30.
b) 3 7  3 9  3 7  9  3 63
x
5
x
5
5
x
c) 4  4  4   4
3 z
3 z
3z
Bruce Mayer, PE
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Caveat Product Rule
 CAUTION
 The product rule for radicals applies
only when radicals have the SAME
index:
n a  m b  nm a  b .
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Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt
Example  Product Rule
 Find the product and write the answer in
simplest form. Assume all variables
represent nonnegative values.
a)
4
b)
7
y5  7 y9
b)
7
y5  7 y9 
7
y5  y9
 4 16

7
y14
2
 y2
24 8
 SOLN
a)
4
2  4 8  4 2 8
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Bruce Mayer, PE
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Example  Product Rule
 Multiply
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 SOLUTION
Bruce Mayer, PE
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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
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Bruce Mayer, PE
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Simplify by Product Rule
 Use The Product Rule in
REVERSE to Facilitate the
Simplification process
n ab  n a  n b
• Note that nnaaand
and
andnnbb must both be
real numbers
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Bruce Mayer, PE
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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.
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Bruce Mayer, PE
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Nix Negative Radicands
 It is often safe to assume that a
radicand does not represent a
negative number when the
radicand is raised to an even power
 To Clarify the Essence of Radical
Simplification We will make this
assumption
• i.e., do NOT Need AbsVal bars
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Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt
Example  Simplify by Factoring
 Simplify by factoring
a.
300
b.
4
8m n
c.
3
54s 4
 SOLUTION
a.
300  100  3
100 is the largest perfectsquare factor of 300.
 100  3
 10 3
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Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt
Example  Simplify by Factoring
 SOLUTION: b.
4
4
8m n
c.
3
54s 4
4
8m n  4  2  m  n
b.
 4m 4  2n
 2m 2 2n
c.
3
4
3
3
54s  27  2  s  s
27s3 is the largest perfect
third-power factor.
3
 27 s3  3 2s
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 3s 3 2s
Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt
WhiteBoard Work
 Problems From
§7.3 Exercise Set
• 14, 18, 20, 32,
38, 56, 96


Estimating Dinosaur Speed (h ≡ hip hgt)
v = [gh(SL/1.8h)2.56]0.5 (Thulborn)

v = 0.25*g0.5*SL1.67*h−1.17 (Alexander)
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Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt
All Done for Today
Running
Dinosaur
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Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt
Example  Simplify by Factoring
 Simplify
 SOLUTION
Cannot be simplified further.
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Chabot Mathematics
Appendix
r  s  r  s r  s 
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Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
–
Chabot College Mathematics
19
Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt
Chabot College Mathematics
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Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt
Graph y = |x|
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 Make T-table
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file =XY_Plot_0211.xls
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Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt
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M55_§JBerland_Graphs_0806.xls
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M55_§JBerland_Graphs_0806.xls
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Bruce Mayer, PE
[email protected] • MTH55_Lec-41_sec_7-3a_Radical_Product_Rule.ppt.ppt
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