Chabot Mathematics
§7.2 Radical
Functions
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Review § 7.1
MTH 55
 Any QUESTIONS About
• §7.1 → Cube & nth Roots
 Any QUESTIONS About HomeWork
• §7.1 → HW-30
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.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.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.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
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As 0.2·0.2·0.2 = 0.008
As 33 = 27 and 43 = 64,
so (3/4)3 = 27/64
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Rational Exponents
 Consider a1/2a1/2. If we still want to
add exponents when multiplying, it
must follow from the Exponent
PRODUCT RULE that
a1/2a1/2 = a1/2 + 1/2, or a1
 Recall  [SomeThing]·[SomeThing] = [SomeThing]2
 This suggests that a1/2 is a
square root of a.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Definition of a1/n
1/ n
a
n
means a .
1/ n
a
n
= a
 When a is NONnegative, n can be any
natural number greater than 1.
When a is negative, n must be odd.
 Note that the denominator of the
exponent becomes the index and the
BASE becomes the RADICAND.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.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 omitted.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Evaluating a1/n
 Evaluate Each Expression
(a)
271/3 =
(b)
3
27
= 3
641/2 =
64
= 8
(c)
–6251/4 =
–
(d)
(–625)1/4
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8
=
4
4
625 = –5
–625
is not a real number because the radicand,
–625, is negative and the index is even.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Caveat on Roots
 CAUTION: Notice the difference between
parts (c) and (d) in the last Example.
 The radical in part (c) is the negative fourth
root of a positive number, while the radical in
part (d) is the principal fourth root of a
negative number, which is NOT a real no.
(c)
–6251/4 =
(d)
(–625)1/4
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–
=
4
4
625 = –5
–625
is not a real number because the radicand,
–625, is negative and the index is even.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Radical Functions
 Given PolyNomial, P,
a RADICAL FUNCTION
Takes this form:
f x   P
n
 Example  Given f(x) = 5 x  8,
Then find f(3).
 SOLUTION
 To find f(3), substitute 3 for x and simplify.
f  3  5  3  8  15  8  7
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Example  Exponent to Radical
 Write an equivalent expression using
RADICAL notation
a)
m
1
3
b)
9x 
1
8 2
c)
xy z 
1
5
2
 SOLUTION
1
3
a)
m  m
b)
9 x 
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c)
3
1
8 2
1
2
xy z 
2
1
5
 xy z
 9 x  9 x  3x
4
4
5
2
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Example  Radical to Exponent
 Write an equivalent expression using
EXPONENT notation
a)
3
b)
4x
4
 SOLUTION
a)
3
4 x  4 x 
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13
5y
x
2
b)
4
14
 5y 
5y

 
x
 x 
2
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Exponent ↔ Index
Base ↔ Radicand
 From the Previous Examples Notice:
1
3
The denominator of the exponent
becomes the index. The base
becomes the radicand.
m  m
3
4 x  4 x 
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3
1 3 The index becomes the
denominator of the exponent.
The radicand becomes the base.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Definition of am/n
 For any natural numbers m and n
(n not 1) and any real number a for
which the radical n a exists,
a
m/n
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 
n
means a
m
n m
, or a .
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Example  am/n Radicals

Rewrite as radicals, then simplify
•

a. 272/3
b. 2433/4
c. 95/2
SOLUTION
a. 272/ 3  (271/ 3 )2  ( 3 27 ) 2  32  9
 33  27
3/ 5
1/ 5 3
243

(243
)  ( 5 243)3
b.
c.
5/ 2
9
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 (9 )  ( 9)  3  243
1/ 2 5
5
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Example  am/n Exponents
3 5
x
 Rewrite with rational a)
exponents
a)
3 5
x
b)

5 3 xy
b)
 SOLUTION
a)
b)
3 5
x x

5 3xy
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


5 3 xy

2
2
5/3
2
  3xy 
2/5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Definition of a−m/n
 For any rational number m/n and
any positive real number a the
NEGATIVE rational exponent:
a
m / n
1
means
a
m/n
.
 That is, am/n and a−m/n are
reciprocals
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Caveat on Negative Exponents
 A negative exponent does not
indicate that the expression in
which it appears is negative; i.e.;
a
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1
 a.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Example  Negative Exponents
 Rewrite with positive exponents, & simplify
• a.
8−2/3
b.
c.  3t 
9−3/2x1/5
 
 2r 
 SOLUTION
2 / 3
a) 8
1

b) 9
3/ 2 1/ 5
x
2/3
8

1
 
38 2
1
1


2 4
2
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 3t 
c)  
 2r 
3/ 4
3/ 4

1
 x1/ 5
93/ 2
1 1/ 5 x1/ 5
 x 
27
27
3/ 4
 2r 
 
 3t 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Example  Speed of Sound
 Many applications translate to radical
equations.
 For example, at a temperature of t
degrees Fahrenheit, sound travels S
feet per second According to the
Formula
S  21.9 5t  2457.
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Example  Speed of Sound
 During orchestra practice, the
temperature of a room was 74 °F. How
fast was the sound of the orchestra
traveling through the room?
S  21.9 5t  2457
 SOLUTION:
Substitute 74 for t in
S  21.9 5(74)2457
the Formula and find
S  21.9 370 2457
an approximation
S  21.9 2827
using a calculator.
ft   3600 s   1 mile 
mile

1164
.
4



793
.
9

s   1 hr   5280 ft 
hr
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S  1164.4 ft/sec.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
WhiteBoard Work
 Problems From §7.2 Exercise Set
• 4, 10, 18, 32, 48, 54, 130

The
MACH No.
M
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BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
All Done for Today
Ernst Mach
Fluid
Dynamicist
 Born 8Feb1838 in
Brno, Austria
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 Died 19Feb1916 (aged
78) in Munich, Germany
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.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
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
ft   3600 s   1 mile 
mile

1164
.
4



793
.
9

s   1 hr   5280 ft 
hr
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
Graph y = |x|
6
 Make T-table
x
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
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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
-5
file =XY_Plot_0211.xls
-6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.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
4
5
-2
-1
-3
-2
M55_§JBerland_Graphs_0806.xls
-3
Chabot College Mathematics
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-4
M55_§JBerland_Graphs_0806.xls
-5
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BMayer@ChabotCollege.edu • MTH55_Lec-39_sec_7-2a_Rational_Exponents.ppt
8
10