Chabot Mathematics
§7.1 Radical
Expressions
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Review § 6.8
MTH 55
 Any QUESTIONS About
• §6.8 → Direct/Indirect Variation &
Modeling
 Any QUESTIONS About HomeWork
• §6.8 → HW-23
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Square Root
The number c is
a square root
2
of a if c = a
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Square Root Examples
 Find the square roots: a) 144 b) 625
 Solution a)
The square roots of 144 are 12 and −12.
To check, note that 122 = 144 and
(−12)2 = (−12)(−12) = 144
 Solution b)
The square roots of 625 are 25 and −25.
To check, note that 252 = 625 and
(−25)2 = (−25)(−25) = 625.
Chabot College Mathematics
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Notation & Nomenclature Notes
 The NONnegative square
root of a number is called
the PRINCIPAL square
root of that number.
 A radical sign, √,
indicates the principal
square root of the number
under the sign
(the radicand).
Chabot College Mathematics
5
Perfect
Squares
Principal
Square Roots
02 = 0
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
102 = 100
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Examples  Principle Sq Roots
 Find the following: a) 100
b)  49
 Soln a) The principal square root of 100
is its positive square root, so 100  10
 Soln b) The symbol  49 represents
the opposite of 49.
 Since
Chabot College Mathematics
6
49  7, we have  49  7.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Expressions of the Form
a
2
 It is tempting to write a  a, but
the next example shows that, as a
rule, this is UNtrue.
 Example 
2
a)
b)
2
8  64  8
(8)  64  8
Chabot College Mathematics
7
2
2
( (8)  8)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Simplifying
a
2
 For any real number a,
a  a.
2
 That is, The principal square root
of a2 is the absolute value of a.
Chabot College Mathematics
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example  SqRt & AbsVal
a) ( y  3) 2
 Simplify each expression. Assume that
2any real
12no.
the variable cana)represent
( y  3) b) m
a)
( y  3)
12
2
m
 b)
SOLUTION
10 2
x
b)
c)
a)c) ( y  3)  y  3
Chabot College Mathematics
9
12
m
c)
10
x
10
x
Since y + 3 might be
negative, absolute-value
notation is necessary.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
( y  3)
a)
Example 
2
12
a)
(
y

3)
b)
m
SqRt & AbsVal
12
 SOLUTION b) & c) b)
m
 SOLUTION b)
x10
c)
• Note that (m6)2 = m12 and that
m6 is NEVER negative. Thus,
x10
c)
12
m
6
m .
 SOLUTION c)
• Note that (x5)2 = x10 and that
x5 MIGHT be negative. Thus
Chabot College Mathematics
10
x10  x5 .
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Plot f(x) = SqRt(x) = √(x)
12
 Plot Using
T-Table
x
0
1
4
9
16
10
8
y = SqRt(x)
Chabot College Mathematics
y  f x   x
6
0
1
2
3
4
 Plot Pts and
Connect with
Smooth Curve
11
y
4
2
x
0
-2
0
2
4
6
8
10
12
-2
-4
-6
-8
M55_§JBerland_Graphs_0806.xls
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
14
16
18
Domain & Range of √x
 Recall that taking the Sq-Root of a
Negative Number does NOT return a
Real-Number Result. Thus the Domain:
{x|x≥0}
 Recall the PRINCIPAL Sq-Root function
return the POSITIVE Root only Thus the
Range for the Principal SqRt fcn:
{y|y≥0}
Chabot College Mathematics
12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Domain & Range of √x
12
 Find Domain &
Range for √x
(the principal
SqRt Fcn) from
the Graph
 Analysis of the
Graph Reveals
y
10
8
y  f x   x
6
4
2
x
0
-2
0
2
4
6
8
10
12
-2
• Domain = {x|x≥0}-4 & Range = {y|y≥0}
 Thus the SqRt Fcn occupies only
the 1st Quadrant of the XY Plane
-6
-8
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
14
16
18
Domain & Range for y  x  3  5
 Need POSITIVE Radicand thus need x ≥ −3
 Also the output of the principal SqRt Fcn is
Always NONnegative so
y
y is at MINIMUM −5
 Thus
8
6
4
• Domain = (−3, )
• Range = (−5, )
2
x
0
-6
-4
-2
0
2
4
6
8
-2
 Graph Confirms D & R
-4
-6
-8
Bruce Mayer, PE
Chabot College Mathematics
14
-10
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
10
12
14
Radicands & Radical Expressions
 A radical expression is an algebraic
expression that contains at least one
radical sign
 Some examples:
24,
Chabot College Mathematics
15
7  3x ,
x  9,
2
y 7
.
3
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Examples  Radicands
 Identify the radicand in expressions:
a) y
b) y 2  6
 Soln a) in
y the radicand is y
 Soln b) in
y 2  6, the radicand is y2 – 6
Chabot College Mathematics
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example  Elliptical Orbit
 When calculating the velocity of a body in
elliptical orbit at a distance r from the focus,
in terms of the SemiMajor axis,
a, we encounter the Expression:
 Evaluate for: r = 10 260
Chabot College Mathematics
17
a = 14 460
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Square Root Functions
 Given a PolyNomial, P, then
a Square-Root Function
f
x
takes the form
 
P
 EXAMPLE  find f(3) for f x   5 x  8
 SOLUTION: To find f(3), substitute 3 for
x and simplify.
f  3  5  3  8  15  8  7
Chabot College Mathematics
18
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Domain of Square Root Fcn
 EXAMPLE  Find the Domain for:
a) f  x   x  8
b) f  x   3x  9
 SOLUTION a) the radicand for a
Sq-Root must be NONnegative thus
x 8  0
x 8
 This InEquality requires this Domain:
 x x  8 , or [8, )
Chabot College Mathematics
19
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Domain of Square Root Fcn
 EXAMPLE  Find the Domain for:
a) f  x   x  8
b) f  x   3x  9
 SOLUTION b) the radicand for a
Sq-Root must be NONnegative thus
3x  9  0
3x  9
x3
 This InEquality requires this Domain:
 x x  3 , or (,3]
Chabot College Mathematics
20
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Industrial Engineering Modeling
 The attendants at a downtown parking
lot use staging-spaces to leave cars
before they are taken to long-term
parking stalls. The required number, N,
of such spaces is approximated by the
formula:
N  2.5 A ,
• where A is the average number of arrivals
during peak hours.
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Industrial Engineering Modeling
 For N  2.5 A ,
 Find the number of spaces needed
when an average of 62 cars arrive
during peak hours
 SOLUTION → Substitute 62 into the
formula and use a calculator to find an
approximation:
Note that we round up to
N  2.5 62
 2.5(7.874)
 19.685  20
Chabot College Mathematics
22
20 spaces because rounding
down would create some
overcrowding.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Simplified Form of a Square Root
 A radical expression for a
square root is simplified when
its radicand has no factor other
than 1 that is a perfect square.

Chabot College Mathematics
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example  Simplification
 Simplify by factoring (note that all
variables are assumed to represent
nonnegative numbers).
a) 24
b) w4 y
c) 600x 2 y
 Soln a)
24  4  6
 4 6
2 6
Chabot College Mathematics
24
 Soln c)
 Soln b)
w4 y  w4 y
 w2 y
600x 2 y
600 x 2 y  100 6 x 2 y
 10 x 6 y
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example  Evaluation
 Evaluate: b2  4ac
• for a = 4, b = 7 and c = −2.
 Solution:
b 2  4ac  7 2  4  4  (2)
 49 16  (2)
 49  32
 81
9
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Simplifying Sq-Roots of Powers
 To take the square root of an EVEN
power such as x12, note that x12 = (x6)2
 Thus
x12  ( x 6 ) 2  x 6
 The exponent of the square root is half
the exponent of the radicand. That is,
x12  x6 .
Chabot College Mathematics
26
1
(12)  6
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example: Sq-Roots of Powers
 Simplify: a) x8
b) x14
c) y 32
 Soln a) x8  ( x 4 )2  x 4
Half of 8 is 4.
 Soln b) x  ( x )  x
Half of 14 is 7.
14
 Soln c)
Chabot College Mathematics
27
7 2
7
y 32  ( y16 )2  x16
Half of 32 is 16.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example: More Powers
 Simplify: a) x 25
 Solution a)
x
25
 x x
24
b) 27x17
 Solution b)
27 x17  9 x16  3x
 x 24  x
 9 x16 3x
 x12 x
 3x8 3x
Caution! The square
root of x16 is not x4.
Chabot College Mathematics
28
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example  Free Fall
 The time (t) it takes in seconds
to fall d feet is given by
t
d
16
 Find the Free-Fall time for an 800ft Drop
 Familiarize: Need to find the time it
takes for an object to fall 800 feet
 Translate: Use the formula, substituting
800ft for d t  800
Replace d with 800.
16
 CarryOut: t  50 Divide within the radical.
t  7.071
Chabot College Mathematics
29
Evaluate the square root.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example  Jump to HyperSpace
 Your sister is 5 years older than you
are. She decides she has had enough
of Earth and needs a vacation. She
takes a trip to the Omega-One star
system. Her trip to Omega-One and
back in a spacecraft traveling at an
average speed v took 15 years,
according to the clock and calendar on
the spacecraft.
Chabot College Mathematics
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example  Jump to HyperSpace
 (cont.) But on landing back on Earth, she
discovers that her voyage took 25 years,
according to the time on Earth. This
means that, although you were 5 years
younger than your sister before her
vacation, you are now 5 years older
than she is after the interstellar vacation!
 Find the StarShip’s
2
v
speed using Einstein’s t 0  t 1  2
c
time-dilation eqn:
Chabot College Mathematics
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
Example  Jump to HyperSpace
 SOLN: Sub t0 = 15 (moving-frame time)
and t = 25 (fixed-frame time) to obtain
2
v
15  25 1  2
c
2
v
3
 1 2
c
5
2
v
9
 1 2
c
25
Chabot College Mathematics
32
v2
9
 1
2
c
25
v
4

c
5
2
4
16
 v
v  c  0.8c
  
5
c
25
 So the StarShip was
moving at 80% the
speed of light (0.8c)
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
WhiteBoard Work
 Problems From §7.1 Exercise Set
• 16, 18, 24,
46, 100

Twins
Encounter
Time
Dilation
Chabot College Mathematics
33
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
All Done for Today
Child BMI
Growth
Chart
Chabot College Mathematics
34
Bruce Mayer, PE
BMayer@ChabotCollege.edu • 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
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
35
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
All Done for Today
SkidMark
Analysis
Skid Distances
Chabot College Mathematics
36
Bruce Mayer, PE
BMayer@ChabotCollege.edu • 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
37
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-37_sec_7-1a_Radical_Expressions.ppt
4
5
6
y
5
12
4
y
10
8
3
6
2
4
1
x
2
0
-3
-2
-1
0
-1
-2
1
2
-2
3
0
0
4
2
5
4
x
6
8
10
12
14
-2
-4
-6
M55_§JBerland_Graphs_0806.xls
-3
-8
M55_§JBerland_Graphs_0806.xls
Chabot College Mathematics
38
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt
16
18
y
6
5
4
3
x
2
1
0
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
-1
-2
-3
-4
-5
M55_§JBerland_Graphs_0806.xls
-6
Chabot College Mathematics
39
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH55_Lec-37_sec_7-1a_Radical_Expressions.ppt