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Roots & Radicals
Intermediate Algebra
1
Roots and Radicals
Radicals
Rational Exponents
Operations with Radicals
Quotients, Powers, etc.
Solving Equations
Complex Numbers
2
Radicals
7.1
3
Square Roots
Finding Square Roots
32
= 9
(-3)2 = 9
(½)2= (¼)
N.B. -32
=
-9
The square root of 9 is 3
The square root of 9 is also –3
The square root of (¼) is (½)
4
Square Roots
The square root symbol
Radical sign
The expression within is the radicand
Square Root
If a is a positive number, then
is the positive square root of a
a
 a is the negative square root of a
Also,
0 0
5
Approximating Square Roots
Approximating Square Roots
Perfect squares are numbers whose square roots
are integers, for example 81 = 92.
Square roots of other numbers are irrational
numbers, for example
2, 3
We can approximate square roots with a
calculator.
6
Approximating Square Roots
10  3.162
(Calculator)
We can determine that it is greater than 3 and less
then 4 because 32 = 9 and 42 =16.
7
Cube Roots
2 is the cube root of 8 because 23 = 8.
3
8 2 2
3
3
8 and 23 above are radicands
3 is called the index (index 2 is omitted)
3
27  ?
3
27
?
8
3
 27  ?
.
8
Cube Roots Evaluated
2 is the cube root of 8 because 23 = 8.
3
8 2 2
3
3
8 and 23 above are radicands
3 is called the index (index 2 is omitted)
3
27  3
3
27 3

8
2
3
27  3
9
nth Roots
n
The number b is an nth root of a, a , if
bn = a.
10
nth Roots
nth Roots
An nth root of number a is a number whose nth
power is a.
n
a  a number whose nth power is a
If the index n is even, then the radicand a must
be nonnegative.
4
16  2, but 4 16 is not a real number
5
32  2
11
age 397
Radicals
12
7-8
age 393
Square Root of x2
x x
2
13
7-7
age 398
Product Rule for Radicals
14
7-9
Simplifying Radical Expressions
Product Rule –
n
a  n b  n a b
36 
4  6  2  12
36 
4 
10 y 
144  12
3k 
30ky
4
2  5 2  XXXXXX
5
9 xy2  5 8 xy2 
5
72 x 2 y 4
15
age 399
Quotient Rule for Radicals
16
7-10
age 399
Quotient Rule for Radicals
64
64 8


49
49 7
 27 x

9
64 y
6
3
17
7-10
Quotient Rule for Radicals
64
64 8


49
49 7
 27 x

9
64 y
6
3
age 399
 27 x
3
3
64 y 9
6

 27  x
3
3
3
64  3 y 9
6

 3
4
3
3
x 
y 
2 3
3 3
 3x 2

4 y3
18
7-10
Radical Functions
Finding the domain of a square root function.
f ( x)  2 x  12
19
Radical Functions
Finding the domain of a square root function.
f ( x)  2 x  12
Domain is all x ' s for which 2 x  12  0.
That is,  x | x  6.
20
Warm-Ups 7.1
21
7.1 T or F
1.
2.
3.
4.
5.
T
F
T
F
T
6. F
7. F
8. F
9. T
10. T
22
Wind Chill
23
Wind Chill
24
Wind Chill
10.5  6.7

W  91.4 

20  0.45  20   457  5  25  
110
25
Wind Chill
10.5  6.7

W  91.4 

20  0.45  20   457  5  25  
110
 3.56
26
Rational Exponents
7.2
27
Exponent 1/n When n Is Even
Page 388
28
7
When n Is Even
1
2
100  100  10
1
4
625  625  5
4
1
6
64  64  2
 4
6
1
2
  4 is not yet defined
29
Exponent 1/n When n Is Odd
Page 389
30
7
Exponent 1/n When n Is Odd
1
3
27  27  3
3
 27 
1
3
  27  3
3
1
5
1
1
 1 

  5
32 2
 32 
31
age 389
nth Root of Zero
0 0 
n
n
0 0
32
age 390
Rational Exponents
33
7-4
Evaluating in Either Order
8
 8   2
2
3

2
3
 8  64  4
2
3
2
4
or
8
3
2
3
34
age 391
Negative Rational Exponents
35
7-5
Evaluating a - m/n
8
2

3

1
8
2
3

1
 8
3
2
1
1
 2 
2 4
36
Rules for Rational Exponents
37
Page 392
7
Simplifying
y 
1
6 6

6
y  y
6


 a b ab  




1
2
1

3
38
Simplifying



 1 1
 a b ab    a b  a b








1
2
1

3
1
2
a
1

3
 
1
1
1  1
3
2
b
3
2
a b
2
3
39
Simplifying
9 x y
8

1
10 12 2
z

40
Simplifying
9 x y
8

1
10 12 2
z
1
2
4 6
3x z
9 x y z 
5
y
4
5 6
41
Simplified Form for Radicals of Index n
A radical expression of index n is in Simplified
Radical Form if it has
1. No perfect nth powers as factors of the radicand,
2. No fractions inside the radical, and
3. No radicals in the denominator.
42
Warm-Ups 7.2
43
7.2 T or F
1.
2.
3.
4.
5.
T
F
F
T
T
6. T
7. T
8. F
9. T
10. T
44
California Growing
1
n
S
r    1
P
45
Growth Rate
1
n
S
r    1
P
1
30
 32.5 
r 
  1  0.0165  1.65%
 19.9 
46
Operations with Radicals
7.3
47
Addition and Subtraction
Like Radicals
3 2 2 2  2 4 2
3 2 2 2  2 4 2
5
5
5
5
48
Addition and Subtraction
Like Radicals
3 22 2 2 4 2
3 22 2 2 4 2
5
5
5
5
but
3 22 3 5 3 22 3 5
49
Simplifying Before Combining
8  18 
50
Simplifying Before Combining
8  18  4 2  9 2
 2 2 3 2
5 2
51
Simplifying Before Combining
1
 20 
5
52
Simplifying Before Combining
1
1
 20 
 4 5
5
5
53
Simplifying Before Combining
1
1
 20 
 4 5
5
5
1 5


2 5
5 5
54
Simplifying Before Combining
1
1
 20 
 4 5
5
5
1 5


2 5
5 5
5

2 5
5
5 10 5 11 5



5
5
5
55
Simplifying Before Combining
1
11 5
 20 
5
5
3
16 x y  54 x y 
4
3
3
4
3
56
Simplifying Before Combining
3
16 x y  54 x y
4
3
4
3
3
 8 2 x  x y
3
3
3
3
3
3
3
 27  2  x  x  y
3
3
3
3
3
3
3
57
Simplifying Before Combining
3
16 x y  54 x y
4
3
4
3
3
 8 2 x  x y
3
3
3
3
3
3
3
 27  2  x  x  y
3
3
3
3
3
3
3
 2 2 x x  y 3 2  x x  y
3
3
3
3
58
Simplifying Before Combining
3
16 x y  54 x y
4
3
4
3
3
 8 2 x  x y
3
3
3
3
3
3
3
 27  2  x  x  y
3
3
3
3
3
3
3
 2 2 x x  y 3 2  x x  y
3
3
3
3
 2 xy 2 x  3xy 2 x
3
  xy 2 x
3
3
59
Multiplying Radicals
Same index
2  5  10
3 2  4 5  12 10
7
x y  x y x y
2
3
7
3
7
5
4
60
Multiplying Radicals
Same index
2  5  10
3 2  4 5  12 10
7
x y  x y x y
2
3
4
3
3
7
7
5
4
2
x 4 x


2
4
61
Multiplying Radicals
Same index
3
4
2
5
x 4 x
x

4

2
4
8
62
Multiplying Radicals
Same index
3
4
2
5
x 4 x
x

4

2
4
8
4
x  x

4
8
4
4
63
Multiplying Radicals
Same index
3
4
2
5
x 4 x
x

4
2
4
8

4
x  x
4
8
4
4
x x
 4

8
4
64
Multiplying Radicals
3
Same index
4
5
2
x
x 4 x
4

8
4
2

4
x  x
4
8
4
4
x x 2
4 
 4
2
8
4
4
65
Multiplying Radicals
Same index
3
4
2
5
x 4 x
x

4
2
4
8
x x 2
 4
4
8
2
4
4
x  2x
 4

16
4
66
Multiplying Radicals
Same index
3
4
2
5
x 4 x
x

4
2
4
8
x4 x 4 2
 4
4
8
2
x  2x
 4
16
4
x  2x

2
4
67
Multiplying Radicals - Binomials
3 2


2  4 5  3  2  12 10  6  12 10
68
Multiplying Binomials
3 2
3
3

2
3

2  4 5  3  2  12 10  6  12 10
2 4
3
5 3
3
4  12 10
3
69
Multiplying Binomials
3 2
3

3 2


2  4 5  3  2  12 10  6  12 10

3

2  4 5  3 4  12 10
24 5
3

3
3

2 4 5 
70
Multiplying Binomials
3 2
3

3 2


2  4 5  3  2  12 10  6  12 10

3
24 5
3 

2  4 5  3 4  12 10
3


3

2  4 5  2  16  5  78
x 3 
2
3
71
Multiplying Binomials
3 2
3

3 2


2  4 5  3  2  12 10  6  12 10

3
24 5
3 

2  4 5  3 4  12 10
3

3
3

2  4 5  2  16  5  78 conjugates

x  3  9  6 x  3  x  3  6  x  6 x  3
2
72
Multiplying Radicals – Different Indices
1
4
1
2
4
2  2  2 2  2
3
2 3 
1 1

4 2
3
4
2  2  8
4
3
4
73
Multiplying Radicals
Different Indices
4
3
1
4
1
2
1
3
1
2
2  2  2 2  2
1 1

4 2
3
4
2  2  8
4
3
4
2  3  2 3 
74
Different Indices
4
3
1
4
1
2
1 1

4 2
1
3
1
2
2
6
2  2  2 2  2
3
4
2  2  8
4
3
4
3
6
2  3  2 3  2 3 
75
Different Indices
4
3
1
4
1
2
1 1

4 2
1
3
1
2
2
6
2  2  2 2  2
3
4
2  2  8
3
6
3
4
4
2  3  2 3  2 3  2  3 
6
2
6
3
76
Different Indices
4
3
1
4
1
2
1 1

4 2
1
3
1
2
2
6
2  2  2 2  2
3
4
2  2  8
3
6
3
4
4
2  3  2 3  2 3  2  3
6
2
6
3
 6 4  6 27
 108
6
77
Conjugates
x  y x  y   x

2 3

2
y
2

2  3  2  3  1
3  2 7 3  2 7  
78
Conjugates
x  y x  y   x

2 3

2
y
2

2  3  2  3  1
3  2 7 3  2 7   9  4  7  19
79
Warm-Ups 7.3
80
7.3 T or F
1.
2.
3.
4.
5.
F
T
F
F
T
6. F
7. T
8. F
9. F
10. T
81
Area of a Triangle
82
Area of a Triangle
1
A  bh
2
83
Area of a Triangle
1
A  bh
2
1
2
A
30 6  3 5 m
2
84
Quotients, Powers, etc
7.4
85
Dividing Radicals
10
10
10  5 

 2
5
5
86
Dividing Radicals
10
10
10  5 

 2
5
5
or
10 5
10  5 


5
5
50
 5
2
5 2

 2
5
87
Dividing Radicals
 6  20  6  2 5

2
2
2 3 5

2
 3  5


88
Rationalizing the Denominator
2 3
2 6
89
Rationalizing the Denominator
2  3   
 2  6 
2 6
2 6


90
Rationalizing the Denominator
2  3   
 2  6 
2 6
2 6


2 2  2 6  6  18

26
91
Rationalizing the Denominator
2  3   
 2  6 


2  6 2 2  2 6  6  18

26
2 6
2 2 3 6 3 2

4
5 2 3 6

4
92
Powers of Radical Expressions
3 3 
4
2 y
3
3
4y
4
 3
4
 81  9  729

3
93
Powers of Radical Expressions
3 3 
4
2 y
3
3

4
3
 3
4
4y  2 y
3
 81 9  729

3 3

3
4 y  8 y  4 y  32 y
3
94
4
Warm-Ups 7.4
95
7.4 T or F
1.
2.
3.
4.
5.
T
T
F
T
F
6. T
7. F
8. T
9. T
10. T
96
7.4 #102

3

 
6 1
3
LCD 

6 1


6 1
6  1


3




 6  1  6  1  6  1 
3  6  1
3  6  1


3
5

6  1
6 1
5
3 6 3
3 6 3


5
5


6 1
3 6 3

5
3 6 3
2
3
5
97
Adding Fractions
110.
x
5

x 3
x
98
Adding Fractions
110.
x
5

x 3
x
LCD 

 x 
x 3
99
Solving Equations
7.5
100
Solving Equations
The Odd Root Property
If n is an odd positive integer,
x k
n

x k
n
for any real number k.
101
Solving Equations – Odd Powers
The Odd Root Property
If n is an odd positive integer,
x k
n

x k
n
for any real number k.
x 8
3
x 82
3
102
Solving Equations – Odd Powers
The Odd Root Property
If n is an odd positive integer,
x k
n

xn k
for any real number k.
x  27
3
x   27  3
3
103
Solving Equations – Odd Powers
The Odd Root Property
If n is an odd positive integer,
x k

n
xn k
for any real number k.
 x  1
3
 54
x  1  54  27  2
3
3
3
x  1 3 2
3
104
age 419
Even-Root Property
105
7-11
Even-Root Property
x  4  x  2
2
x 0 x0
2
x   4 has no real solution
age 419
2
106
7-11
Solving Equations – Even Powers
The Even Root Property
If n is an even positive integer,
k 0

xn  k

x  n k
k 0

xn  k

x0
k 0

xn  k
has no real solution.
x2  4
x   4  2
2, 2
107
Solving Equations – Even Powers
The Even Root Property
If n is an even positive integer,
k 0

xn  k

x  n k
k 0

xn  k

x0
k 0

xn  k
has no real solution.
x 4  1  80
x 4  81
4
x 4   4 81
x  3
3,3
CHECK
108
Solving Equations – Even Powers
The Even Root Property
If n is an even positive integer,
k 0

xn  k

x  n k
k 0

xn  k

x0
k 0

xn  k
has no real solution.
 x  3
2
4
 x  3
2
 4
x  3  2
x  3 2
109
Solving Equations – Even Powers
The Even Root Property
If n is an even positive integer,
k 0

x k

x k
k 0

xn  k

x0
k 0

xn  k
has no real solution.
n
 x  3
2
n
4
x  3 2
x  3 2  5
x  3 2 1
1,5
CHECK
110
Isolating the Radical
2x  3  5  0

2x  3  5

2
2x  3  5
Isolate the radical
2
Square both sides
2 x  3  25
111
Squaring Both Sides
2x  3  5  0
2x  3  5

2x  3

2
Isolate the radical
5
2
Square both sides
2 x  3  25
2 x  28
x  14
14
CHECK
112
Cubing Both Sides
3

a  3  2a  7
3
3
a3
 
3
3
2a  7

3
a  3  2a  7
a  10
10
CHECK
113
Squaring Both Sides Twice
2x 1  x  1

2x 1  x 1
 
2
2x 1 

x 1
2
114
Squaring Both Sides Twice
2x 1  x  1

2x 1  x 1
 
2
2x 1 

x 1
2
2x 1  x  2 x 1
x2 x
 
2
x  2 x  4x
2
115
Squaring Both Sides Twice

2x 1  x  1
x  4x
2x 1  x 1
x  4x  0
 
2
2x 1 
2
2

x 1
2x 1  x  2 x 1
x2 x
 
2
x  2 x  4x
2
2
x  x  4  0
x0
x4
0, 4
CHECK
116
Rational Exponents
Eliminate the root, then the power
2
3
a 2
117
Eliminate the Root, Then the Power
2
3
a 2
3
 
3
a   2
 
2
a 8
2
3
a  8
2
a  2 2
CHECK
2
2, 2 2

118
Negative Exponents
r  1
2

3
1
119
Negative Exponents
Eliminate the root, then the power
 r  1

2
3
1
3


3
r

1

1
 



2

3
 r  1
2
1
 r  1
2
 1
r  1  1
r2
CHECK
r 0
0, 2
120
Negative Exponents
Eliminate the root, then the power
2t  3

2
3
 1
121
No Solution
Eliminate the root, then the power
2t  3

2
3
 1
3
 2t  3    13




2

3
2t  3  1
2
2t  3  
2
1
122
No Solution
Eliminate the root, then the power
 2t  3

2
3
 1
3
3


  2t  3    1


2

3
 2t  3
2
 1
 2t  3
2
  1

No real solution
123
Strategy for Solving Equations with
Exponents and Radicals
Page 424
124
7-1
Distance Formula
Pythagorean Theorem
age 424
a2 + b2 = c2
125
7-13
Distance Formula
Find the distance between the points (-2,3) and (1, -4).
age 424
d
1   2   4  3
2
2
126
7-13
Distance Formula
Find the distance between the points (-2,3) and (1,-4).
age 424
d
d
1   2   4  3
2
2
3   7 
2
d  9  49  58
2
127
7-13
Diagonal of a Sign
What is the length of the diagonal of a rectangular billboard
whose sides are 5 meters and 12 meters?
Let x  diagonal length
a b  c
2
2
2
128
Diagonal of a Sign
What is the length of the diagonal of a rectangular billboard
whose sides are 5 meters and 12 meters?
Let x  diagonal length
a b  c
2
2
2
5  12  x
2
2
2
25  144  x 2
x  169
2
129
Diagonal of a Sign
What is the length of the diagonal of a rectangular billboard whose
sides are 5 meters and 12 meters?
Let x  diagonal length
a 2  b2  c2
52  122  x 2
25  144  x 2
x 2  169
x   169
2
x  13
or
13
The diagonal is 13 meters.
CHECK
130
Warm-Ups 7.5
131
7.5 T or F
1.
2.
3.
4.
5.
F
T
F
F
T
6. F
7. F
8. T
9. T
10. T
132
Complex Numbers
7.6
133
Imaginary Numbers
Define i  1 ,  i   1 ,
2
b  i b
134
Imaginary Numbers
Define i  1 ,  i   1 ,  b  i b
2
 100  i 100  10i
  81  ?
135
Imaginary Numbers
Define i  1 ,  i   1 ,  b  i b
2
 100  i 100  10i
  81   i 81  9i
136
Imaginary Numbers
Define
b  i b
i  1 ,  i  1 ,
2
Beware
4  9  i 4  i 9  i  2  i  3  6i  6  1  6
2
4  9 
 4  9  
36  6
137
Imaginary Numbers
Define
i  1 ,  i  1 ,
2
b  i b
2  8  ?
138
Imaginary Numbers
Define
i  1 ,  i   1 ,
2
b  i b
 2   8  i 2 i 8
i
2
 4i
16
2
 4 1
 4
139
Powers of i
i  i  1
1
i  1
2
i  i  i   1i  i
3
2
i  i  i   1 1  1
4
2
2
i  i  i  1i  i
5
etc.
4
1
140
age 429
Complex Numbers
141
7-14
age 430 (Figure 7.3)
Figure 7.3
142
7-15
Addition and Subtraction
The sum and difference a + bi of c + di and are:
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi) - (c + di) = (a - c) + (b - d)i
143
(2 + 3i) + (4 + 5i)
The sum and difference a + bi of c + di and are:
(2 + 3i) + (4 + 5i) = (2 + 4) + (3 + 5)i
= 6 + 8i
(2 + 3i) – (4 + 5i) = (2 – 4) + (3 – 5)i
= – 2 – 2i
144
Multiplication
The complex numbers a + bi of c + di and are
multiplied as follows:
(a + bi) (c + di) = ac + adi + bci + bdi2
= ac + bd(– 1) + adi + bci
= (ac – bd) + (ad + bc)i
145
(2 + 3i) (4 + 5i)
The complex numbers a + bi of c + di and are
multiplied as follows:
(a + bi) (c + di) = (ac – bd) + (ad + bc)i
(2 + 3i) (4 + 5i) = 8 + 10i + 12i + 15i2
= 8 + 22i + 15(– 1)
= – 7 + 22i
146
Division
(2 + 3i) ÷ 4 = (2 + 3i) / 4
=½+¾i
147
Complex Conjugates
The complex numbers a + bi and a – bi are
called complex conjugates. Their product
is a2 + b2.
148
Division
We divide the complex number a + bi by the
complex number c + di as follows:
a  bi a  bi c  di 

c  di c  di c  di 
149
Division
We divide the complex number a + bi by the
complex number c + di as follows:
2  3i
4  5i
150
Division
We divide the complex number 2 + 3i by the
complex number 4 + 5i.
2  3i 2  3i 4  5i 

4  5i 4  5i 4  5i 
8  10i  12i  15i

2
16  25i
23  2i 23  2i 23 2



 i
16  25
41
41 41
2
151
Square Root of a Negative Number
For any positive real number b,
 b  i b.
152
Imaginary Solutions to Equations
x  25
2
3x  4  0
2
153
Complex Numbers
1.
2.
3.
4.
5.
6.
7.
8.
Definition of i: i =
, i2 = -1.
Complex number form: a + bi.
a + 0i is the real number a.
b is a positive real number   b  i b .
The numbers a + bi and a - bi are complex conjugates. Their
product is a2 + b2.
Add, subtract, and multiply complex numbers as if they
were algebraic expressions with i being the variable, and
replace i2 by -1.
Divide complex numbers by multiplying the numerator and
denominator by the conjugate of the denominator.
In the complex number system x2 = k for any real number k
is equivalent to x   k .
154
Complex Numbers
155
Warm-Ups 7.6
156
7.6 T or F
1.
2.
3.
4.
5.
T
F
F
T
T
6. T
7. T
8. F
9. T
10. F
157
158
159