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Chabot Mathematics. §7.1 Cube & n th Roots. Bruce Mayer, PE Licensed Electrical & Mechanical Engineer [email protected]. MTH 55. 7.1. Review §. Any QUESTIONS About §7.1 → Square-Roots and Radical Expressions Any QUESTIONS About HomeWork §7.1 → HW-24. Cube Root. - PowerPoint PPT Presentation
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[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§7.1 Cube§7.1 Cube& n& nthth Roots Roots
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt2
Bruce Mayer, PE Chabot College Mathematics
Review §Review §
Any QUESTIONS About• §7.1 → Square-Roots and Radical
Expressions
Any QUESTIONS About HomeWork• §7.1 → HW-24
7.1 MTH 55
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt3
Bruce Mayer, PE Chabot College Mathematics
Cube RootCube Root
The CUBE root, c, of a Number a is written as:
The number c is the cube root of a, if the third power of c is a; that is; if c3 = a, then
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt4
Bruce Mayer, PE Chabot College Mathematics
Example Example Cube Root of No.s Cube Root of No.s
Find Cube Rootsa) b) c)3 0.008 3 27
643 2197
SOLUTION• a) As 0.2·0.2·0.2 = 0.008
• b) 1321973 As (−13)(−13)(−13) = −2197
• c) As 33 = 27 and 43 = 64,so (3/4)3 = 27/64
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt5
Bruce Mayer, PE Chabot College Mathematics
Cube Root FunctionsCube Root Functions
Since EVERY Real Number has a Cube Root Define the Cube Root Function:
3 xxf The Graph
Reveals• Domain =
{all Real numbers}
• Range ={all Real numbers}
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M55_§JBerland_Graphs_0806.xls
x
y
x
y
3 xxfy
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt6
Bruce Mayer, PE Chabot College Mathematics
Evaluate Cube Root FunctionsEvaluate Cube Root Functions
Evaluate Cube Root Functionsa)
b)
731923 uyyu
173773 vzzv
SOLUTION (using calculator)a)
b)
0255127
1914619732733
33
.
u
344482
3711937177173
33
.
v
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt7
Bruce Mayer, PE Chabot College Mathematics
Simplify Cube RootsSimplify Cube Roots
For any Real Number, a aa 3 3
Use this property to simplify Cube Root Expressions.
For EXAMPLE Simplify
SOLUTION
because (–3x)(–3x)(–3x) = –27x3
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt8
Bruce Mayer, PE Chabot College Mathematics
nnthth Roots Roots
nth root: The number c is an nth root of a number a if cn = a.
The fourth root of a number a is the number c for which c4 = a. We write for the nth root. The number n is called the index (plural, indices). When the index is 2 (for a Square Root), the Index is ommitted.
n a
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt9
Bruce Mayer, PE Chabot College Mathematics
Odd & Even Odd & Even nnthth Roots → Roots →
When the index number, n, is ODD the root itself is also called ODD• A Cube-Root (n = 3) is Odd. Other Odd
roots share the properties of Cube-Roots– the most important property of ODD roots is
that we can take the ODD-Root of any Real Number – positive or NEGATIVE
– Domain of Odd Roots = (−, +)
– Range of Odd Roots =(−, +)
n a
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt10
Bruce Mayer, PE Chabot College Mathematics
Example Example nnthth Roots of No.s Roots of No.s
Find ODD Rootsa) b) c)
SOLUTION• a) Since 35 = 243
• b)
5 243
As (−3)(−3)(−3)(−3)(−3) = −243
• c)When the index equals the exponent under the radical we recover the Base
5 243 11 11m
32435
32435
mm 11 11
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt11
Bruce Mayer, PE Chabot College Mathematics
Odd & Even Odd & Even nnthth Roots → Roots →
When the index number, n, is EVEN the root itself is also called EVEN• A Sq-Root (n = 2) is Even. Other Even
roots share the properties of Sq-Roots– The most important property of EVEN roots is
that we canNOT take the EVEN-Root of a NEGATIVE number.
– Domain of Even Roots = {x|x ≥ 0}
– Range of Even Roots = {y|y ≥ 0}
n a
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt12
Bruce Mayer, PE Chabot College Mathematics
Example Example nnthth Roots of No.s Roots of No.s
Find EVEN Rootsa) b) c)
SOLUTION• a) Since 34 = 81
• b)
4 81
Even Root is Not a Real No.
• c)Use absolute-value notation since m could represent a negative number
4 81 4 416m
3814
4 81
mm 2164 4
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt13
Bruce Mayer, PE Chabot College Mathematics
Simplifying Simplifying nnthth Roots Roots
n a
Even
Positive Positive a
Negative Not a real number
|a|
Odd
Positive Positive a
Negative Negative a
n a nn a
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt14
Bruce Mayer, PE Chabot College Mathematics
Example Example Radical Radical ExpressionsExpressions Find nth Roots
a) b) c)
SOLUTION• a)
• b)
4 42 7u
• c)
5 5113 v 6 613
EVEN is if as 774 4 naauu n n
ODD is if as 1131135 5 naavv n n
EVEN is if as 1313136 6 naan n
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt15
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard WorkWhiteBoard Work
Problems From §7.1 Exercise Set• 50, 74, 84, 88, 98, 102
Principalnth Root
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt16
Bruce Mayer, PE Chabot College Mathematics
All Done for TodayAll Done for Today
SkidMarkAnalysis
Skid Distances
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt17
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
AppendiAppendixx
–
srsrsr 22
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt18
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
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-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
file =XY_Plot_0211.xls
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt19
Bruce Mayer, PE Chabot College Mathematics
y
-16
-14
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0
2
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-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2
M55_§JBerland_Graphs_0806.xls
x
xy