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2
Roots and Radicals
Radicals
Rational Exponents
Operations with Radicals
Quotients, Powers, etc.
Solving Equations
Complex Numbers
4
Square Roots
Finding Square Roots
32 = 9(-3)2 = 9 N.B. -32 = -9(½)2= (¼)
The square root of 9 is 3The square root of 9 is also –3The square root of (¼) is (½)
5
Square RootsThe 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
is the negative square root of a
Also, 00 a
a
6
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
We can approximate square roots with a calculator.
3,2
7
Approximating Square Roots
3.162 (Calculator)
We can determine that it is greater than 3 and less then 4 because 32 = 9 and 42 =16.
10
8
Cube Roots
2 is the cube root of 8 because 23 = 8.
8 and 23 above are radicands
3 is called the index (index 2 is omitted)
.
3 33 8 2 2
?27
?8
27
?27
3
3
3
9
Cube Roots Evaluated
2 is the cube root of 8 because 23 = 8.
8 and 23 above are radicands
3 is called the index (index 2 is omitted)
3 33 8 2 2
3
3
3
27 3
27 3
8 2
27 3
11
nth Rootsnth Roots
An nth root of number a is a number whose nth power is a.
a number whose nth power is a
If the index n is even, then the radicand a must be nonnegative.
is not a real number
n a
4 4
5
16 2, 16
32 2
but
15
Simplifying Radical Expressions
Product Rule –
nnn baba
5 425 25 2
54
7289
22
30310
12144436
1226436
yxxyxy
XXXXXX
kyky
18
Quotient Rule for Radicals
7-10Page 399
3
2
3 33
3 32
3 93
3 63
3 9
3 6
39
6
4
3
4
3
64
27
64
27
64
27
7
8
49
64
49
64
y
x
y
x
y
x
y
x
y
x
20
Radical Functions
Finding the domain of a square root function.
( ) 2 12
' 2 12 0.
, | 6 .
f x x
Domain is all x s for which x
That is x x
42
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.
58
Simplifying Before Combining
yxxyxx
yxx
yxx
yxyx
3333
3 333 333
3 333 333
3 343 34
2322
227
28
5416
59
Simplifying Before Combining
3
33
3333
3 333 333
3 333 333
3 343 34
2
2322
2322
227
28
5416
xxy
xxyxxy
yxxyxx
yxx
yxx
yxyx
72
Multiplying Binomials
366336933
785162542542
10124354223
1012610122354223
2
33333
xxxxx
conjugates
97
7.4 #102
3 36 1 6 1
6 1 6 1
6 1 6 13 3
6 1 6 1 6 1 6 1
3 6 1 3 6 1
5 5
3 6 3 3 6 3
5 5
3 6 3 3 6 3 2 3
5 5
LCD
101
Solving Equations
The Odd Root Property
If n is an odd positive integer,
for any real number k.
nn kxkx
102
Solving Equations – Odd Powers
The Odd Root Property
If n is an odd positive integer,
for any real number k.
nn kxkx
28
83
3
x
x
103
Solving Equations – Odd Powers
The Odd Root Property
If n is an odd positive integer,
for any real number k.
nn kxkx
327
273
3
x
x
104
Solving Equations – Odd Powers
The Odd Root Property
If n is an odd positive integer,
for any real number k.
nn kxkx
3
33 3
3
1 54
1 54 27 2
1 3 2
x
x
x
107
Solving Equations – Even PowersThe Even Root Property
If n is an even positive integer,
2
0
0 0
0 .
4
4 2 2,2
n n
n
n
k x k x k
k x k x
k x k has no real solution
x
x
108
Solving Equations – Even PowersThe Even Root Property
If n is an even positive integer,
4
4
44 4
0
0 0
0 .
1 80
81
81
3 3,3
n n
n
n
k x k x k
k x k x
k x k has no real solution
x
x
x
x CHECK
109
Solving Equations – Even PowersThe Even Root Property
If n is an even positive integer,
2
2
0
0 0
0 .
3 4
3 4
3 2
3 2
n n
n
n
k x k x k
k x k x
k x k has no real solution
x
x
x
x
110
Solving Equations – Even PowersThe Even Root Property
If n is an even positive integer,
2
0
0 0
0 .
3 4
3 2
3 2 5 3 2 1 1,5
n n
n
n
k x k x k
k x k x
k x k has no real solution
x
x
x x CHECK
112
Squaring Both Sides
22
2 3 5 0
2 3 5
2 3 5
2 3 25
2 28
14 14
x
x Isolate the radical
x Square both sides
x
x
x CHECK
116
Squaring Both Sides Twice
xxx
xx
xxx
xx
xx
xx
42
2
1212
112
112
112
22
22
2
2
4
4 0
4 0
0 4
0,4
x x
x x
x x
x x
CHECK
120
Negative ExponentsEliminate the root, then the power
2
3
3233
2
2
1 1
1 1
1 1
1 1
1 1
2 0 0,2
r
r
r
r
r
r r
CHECK
123
No SolutionEliminate the root, then the power
2
3
323
3
2
2
2 3 1
2 3 1
2 3 1
2 3 1
t
t
t
t
No real solution
126
Distance Formula
7-13Page 424
Find the distance between the points (-2,3) and (1, -4).
22 3421 d
127
Distance Formula
7-13Page 424
Find the distance between the points (-2,3) and (1,-4).
58499
73
3421
22
22
d
d
d
128
Diagonal of a SignWhat is the length of the diagonal of a rectangular
billboard whose sides are 5 meters and 12 meters?
222 cba
lengthdiagonalxLet
129
Diagonal of a SignWhat is the length of the diagonal of a rectangular
billboard whose sides are 5 meters and 12 meters?
169
14425
125
2
2
222
222
x
x
x
cba
lengthdiagonalxLet
130
Diagonal of a SignWhat is the length of the diagonal of a rectangular billboard whose
sides are 5 meters and 12 meters?
2 2 2
2 2 2
2
2
2
5 12
25 144
169
169
13 13
13 .
Let x diagonal length
a b c
x
x
x
x
x or CHECK
The diagonal is meters
137
Imaginary Numbers
2
2
1 , 1 ,
4 9 4 9 2 3 6 6 1 6
4 9 4 9 36 6
Beware
Define i i b i b
i i i i i
143
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
144
(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
145
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
146
(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
148
Complex Conjugates
The complex numbers a + bi and a – bi are called complex conjugates. Their product is a2 + b2.
149
Division
We divide the complex number a + bi by the complex number c + di as follows:
dicdic
dicbia
dic
bia
151
DivisionWe divide the complex number 2 + 3i by the
complex number 4 + 5i.
iii
i
iii
ii
ii
i
i
41
2
41
23
41
223
2516
2232516
1512108
5454
5432
54
32
2
2
154
Complex Numbers1. Definition of i: i = , i2 = -1.
2. Complex number form: a + bi.
3. a + 0i is the real number a.
4. b is a positive real number
5. The numbers a + bi and a - bi are complex conjugates. Their product is a2 + b2.
6. Add, subtract, and multiply complex numbers as if they were algebraic expressions with i being the variable, and replace i2 by -1.
7. Divide complex numbers by multiplying the numerator and denominator by the conjugate of the denominator.
8. In the complex number system x2 = k for any real number k is equivalent to
.bib
.kx
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