Download MTH55_Lec-38_sec_7-1b_Cube_nth_Roots

Chabot Mathematics
§7.1 Cube
th
& n Roots
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
Chabot College Mathematics
1
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Review § 7.1
MTH 55
 Any QUESTIONS About
• §7.1 → Square-Roots and Radical
Expessions
 Any QUESTIONS About HomeWork
• §7.1 → HW-30
Chabot College Mathematics
2
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Cube Root
 The CUBE root, c, of a
Number a is written as:
3 a,
 The number c is the cube root of a,
if the third power of c is a; that is;
if c3 = a, then
3 a  c.
Chabot College Mathematics
3
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example  Cube Root of No.s
 Find Cube Roots
a) 3 0.008
b)3
27
c)
3
 2197
64
 SOLUTION
• a)
b. 3 0.008  0.2
• b)
3
 2197  13 As (−13)(−13)(−13) = −2197
27
3
c. 3

• c)
64 4
Chabot College Mathematics
4
As 0.2·0.2·0.2 = 0.008
As 33 = 27 and 43 = 64,
so (3/4)3 = 27/64
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Cube Root Functions
 Since EVERY Real Number has a Cube
Root Define the Cube Root Function:
f x   x
5
3
4
2
1
• Domain =
{all Real numbers}
x
0
-10
-8
-6
-4
-2
0
2
4
6
-1
-2
• Range =
{all Real numbers}
-3
-4
M55_§JBerland_Graphs_0806.xls
5
y  f x   3 x
3
 The Graph
Reveals
Chabot College Mathematics
y
-5
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
8
10
Evaluate Cube Root Functions

Evaluate Cube Root Functions
a) u y   3  2 y  19 u73
b) vz   3 7 z  37 v17 

SOLUTION (using calculator)
a) u 73  3  273  19  3  146  19
 3  127  5.025
3 717   37  3 119  37


v
17

b)
 3 82  4.344
Chabot College Mathematics
6
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Simplify Cube Roots
 For any
Real Number, a
3
a a
3
 Use this property to simplify Cube Root
Expressions.
 For EXAMPLE  Simplify 3 27 x3 .
 SOLUTION
3
3
a. 27 x  3 x
Chabot College Mathematics
7
because (–3x)(–3x)(–3x) = –27x3
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
nth 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 n a
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.
Chabot College Mathematics
8
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Odd & Even nth Roots →
n
a
 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 =(−, +)
Chabot College Mathematics
9
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example  nth Roots of No.s
 Find ODD Roots
a)
5
243
b)
5
 243 c)
11
11
m
 SOLUTION
• a)
5
243  3
Since 35 = 243
• b)
5
 243  3
As (−3)(−3)(−3)(−3)(−3) = −243
m m
When the index equals the
exponent under the radical
we recover the Base
• c)
11
Chabot College Mathematics
10
11
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Odd & Even nth Roots →
n
a
 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}
Chabot College Mathematics
11
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example  nth Roots of No.s
 Find EVEN Roots
a)
4
b)
81
4
 81
c)4
16m
4
 SOLUTION
• a)
4
81  3
Since 34 = 81
• b)
4
 81  
Even Root is Not a Real No.
• c)
4
16m  2 m
Chabot College Mathematics
12
4
Use absolute-value notation
since m could represent a
negative number
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Simplifying nth Roots
n
a
n
a
n
n
a
a
Positive
Positive
Negative
|a|
Positive
Not a real
number
Positive
Negative
Negative
a
Even
a
Odd
Chabot College Mathematics
13
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example  Radical Expressions
 Find nth Roots
a) 4 u 2  7 4
b) 5 3v  115 c) 6  136
 SOLUTION
• a)
4
u  7
• b)
5
3v 11
• c)
6
 13
6
n
a n  a if n is EVEN
 3v  11 as
n
a n  a if n is ODD
  13  13 as
n
a n  a if n is EVEN
5
Chabot College Mathematics
14
 u  7 as
4
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
WhiteBoard Work
 Problems From §7.1 Exercise Set
• 50, 74, 84, 88, 98, 102

Principal
nth Root
Chabot College Mathematics
15
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
All Done for Today
SkidMark
Analysis
Skid Distances
Chabot College Mathematics
16
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
[email protected]
–
Chabot College Mathematics
17
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.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
18
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
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
4
5
6
y
4
y
2
x
0
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
2
-2
-4
-6
x
-8
-10
-12
-14
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
19
-16
Bruce Mayer, PE
[email protected] • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt