Download 7.1

Chapter 7
Roots, Radicals,
and Complex
Numbers
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-1
1
Chapter Sections
7.1 – Roots and Radicals
7.2 – Rational Exponents
7.3 – Simplifying Radicals
7.4 – Adding, Subtracting, and Multiplying
Radicals
7.5 – Dividing Radicals
7.6 – Solving Radical Equations
7.7 – Complex Numbers
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-2
2
§ 7.1
Roots and Radicals
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-3
3
Definitions
The symbol √ is called the radical sign and is present
in all radical expressions. The expression under the
radical sign is called the radicand. The number
directly to the left of the radical sign is called the
index and gives us the “root” of the expression.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-4
4
Find Square Roots
A square root is a radical expression that has
an index of 2. The index of a square root is
generally not written. Thus,
x means 2 x
Square Roots
For any positive real number a,
• The principal or positive square root of a, written a , is
the positive number b such that b2 = a.
• The negative square root of a, written - a , is the
opposite of the principal square root of a.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-5
5
Find Square Roots
In this book, whenever we use the words
“square root” , we are referring to the
principal or positive square root.
Note: The square root of a negative number
is not a real number.
 25  ?
There is no number multiplied by itself
that will give you –25.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-6
6
Finding Square Roots
Example For each function, find the indicated
value(s).
a) f ( x)  11x  2, f (6)
f (6)  11(6)  2
 64
8
b) g (r )    3r  1, g(-5)
g (5)    3(5)  1
  16
 4
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-7
7
Find Cube Roots
Cube Root
The cube root of a number a, written a , is the number
b such that b3 = a.
3
Examples
3
82
since 23  8
3
 27  3
since (-3)3  27
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-8
8
Find Cube Roots
Example For each function, find the indicated
value(s).
a) f ( x)  3 10 x  34, f (3)
f (3)  3 10(3)  34
 3 64
4
b) g (r )  3 12r  20, g(-4)
g (4)  3 12(4)  20
 3  68
 4.081655102
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-9
9
Understand Odd and Even Roots
Even Root
The nth root of a, written a , where n is an even index and
a is a nonnegative real number, is called an even root and
is the nonnegative real number b such that bn = a.
n
Examples
93
since 32  3  3  9
4
16  2
since (2)4  2  2  2  2  16
6
729  3
since (3)6  3  3  3  3  3  3  729
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-10
10
Understand Odd and Even Roots
Odd Root
The nth root of a, a , where n is an odd index and a is any
real number, is called an odd root and is the real number b
such that bn = a.
n
Examples
3
82
since 23  2  2  2  8
3
 8  2
5
243  3
since (-2)3  (2)(2)(2)  8
since (3)5  3  3  3  3  3  243
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-11
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Understand Odd and Even Roots
Example Indicate whether or not each radical
expression is a real number. If the expression is a
real number, find its value.
a) 4  81
Not a real number. Even roots of negative numbers are not
real numbers
b) - 4 81
Real number, - 4 81  4 81  (3)  3
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-12
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Evaluate Radicals Using Absolute Value
Radicals and Absolute Value
For any real number a,
a a
2
Examples
92  9  9
02  0  9
(15.7) 2   15.7  15.7
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-13
13
Evaluate Radicals Using Absolute Value
Example Use the absolute value to evaluate.
a)
92  9  9
b)
02  0  0
c)
(15.7) 2   15.7  15.7
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
Chapter 7-14
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