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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
[email protected]
Chabot Mathematics
§7.3 Radical§7.3 RadicalProductsProducts
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Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §7.2 → Rational Exponents
Any QUESTIONS About HomeWork• §7.2 → HW-31
7.2 MTH 55
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Bruce Mayer, PE Chabot College Mathematics
Multiplying Radical ExpressionsMultiplying 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 Chabot College Mathematics
Product Rule for RadicalsProduct Rule for Radicals
For any real numbers and
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 Chabot College Mathematics
Derive Product Rule for RadsDerive Product Rule for Rads
Rational exponents can be used to derive the Product Rule for Radicals:
1/1/ 1/ .nn nn n na b a b a b a b
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Bruce Mayer, PE Chabot College Mathematics
Example Example Product Rule Product Rule
Multiply
a) 5 6
44 5c)
3
x
z
3 3b) 7 9
SOLUTION
3 3 3 3b) 7 9 7 9 63
44 4 45 5 5c)
3 3 3
x x x
z z z
a) 5 6 5 6 30.
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Bruce Mayer, PE Chabot College Mathematics
Caveat Product RuleCaveat Product Rule
CAUTIONCAUTION The product rule for radicals applies
only when radicals have the SAME index:
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Bruce Mayer, PE Chabot College Mathematics
Example Example Product Rule Product Rule
Find the product and write the answer in simplest form. Assume all variables represent nonnegative values.a) b)4 42 8 5 97 7y y
SOLNa) b)4 42 8 4 2 8
4 16
2
5 97 7y y 5 97 y y
147 y
2y
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Bruce Mayer, PE Chabot College Mathematics
Example Example Product Rule Product Rule
Multiply SOLUTION
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Bruce Mayer, PE Chabot College Mathematics
Simplifying by FactoringSimplifying 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 whenever ab contains a factor that is a perfect nth power
n ab
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Bruce Mayer, PE Chabot College Mathematics
Simplify by Product RuleSimplify by Product Rule
Use The Product Rule in REVERSE to Facilitate the Simplification process
• Note that and must both be real numbers
n n nab a b
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Bruce Mayer, PE Chabot College Mathematics
Simplify a Radical Expression Simplify a Radical Expression with Index with Index nn by Factoring 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 Chabot College Mathematics
Nix Negative RadicandsNix 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 Chabot College Mathematics
Example Example Simplify by Factoring Simplify by Factoring
Simplify by factoring
a. 300 4b. 8m n 3 4c. 54s
SOLUTION
a. 300 3100
100 3
10 3
100 is the largest perfect-square factor of 300.
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Bruce Mayer, PE Chabot College Mathematics
Example Example Simplify by Factoring Simplify by Factoring
SOLUTION: 4b. 8m n 3 4c. 54s
4 4b. 8 4 2m n m n
44 2m n 22 2m n
3 34 3c. 54 27 2s s s
3 3 327 2s s 33 2s s
27s3 is the largest perfect third-power factor.
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Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard 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 Chabot College Mathematics
All Done for TodayAll Done for Today
RunningDinosaur
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Bruce Mayer, PE Chabot College Mathematics
Example Example Simplify by Simplify by FactoringFactoring Simplify SOLUTION
Cannot be simplified further.
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Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
[email protected]
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22
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Bruce Mayer, PE Chabot College Mathematics
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Bruce Mayer, PE Chabot College Mathematics
Graph Graph yy = | = |xx||
Make T-tablex y = |x |
-6 6-5 5-4 4-3 3-2 2-1 10 01 12 23 34 45 56 6
x
y
-6
-5
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0
1
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-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
file =XY_Plot_0211.xls
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Bruce Mayer, PE Chabot College Mathematics
-3
-2
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0
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M55_§JBerland_Graphs_0806.xls -5
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M55_§JBerland_Graphs_0806.xls
x
y