Download Square Root

7
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
Exponents and Radicals
Radical Expressions and Functions
Rational Numbers as Exponents
Multiplying Radical Expressions
Dividing Radical Expressions
Expressions Containing Several Radical
Terms
Solving Radical Equations
Geometric Applications
The Complex Numbers
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
7.1

Radical Expressions and
Functions
Square Roots and Square-Root Functions

Expressions of the Form

Cube Roots

Odd and Even nth Roots
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
a
2
Square Roots and SquareRoot Functions
When a number is multiplied by itself,
we say that the number is squared.
Often we need to know what number
was squared in order to produce some
value a. If such a number can be found,
we call that number a square root of a.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 3
Square Root
The number c is a square root of a if c 2 = a.
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Slide 7- 4
For example,
16 has –4 and 4 as square roots because
(–4)2 = 16 and 42 = 16.
–9 does not have a real-number square
root because there is no real number c
for which c 2 = –9.
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 5
Example
Find two square roots of 49.
Solution
The square roots are 7 and –7, because 72 =49
and (–7)2 = 49.
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Slide 7- 6
Whenever we refer to the square root of
a number, we mean the nonnegative
square root of that number. This is often
referred to as the principal square root
of the number.
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Slide 7- 7
Principal Square Root
The principal square root of a nonnegative
number is its nonnegative square root. The
symbol
is called a radical sign and is
used to indicate the principal square root of
the number over which it appears.
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Slide 7- 8
Example
Simplify each of the following.
a)
81
16
b) 
81
Solution
a)
81  9
16
4
b) 

81
9
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Slide 7- 9
a is also read as “the square root of a,”
“root a,” or “radical a.” Any expression
in which a radical sign appears is called a
radical expression. The following are
examples of radical expressions:
12,
3m  2, and
3x  2 x
 3.
7
2
The expression under the radical sign is
called the radicand.
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Slide 7- 10
Expressions of the Form a
2
It is tempting to write a 2  a, but the next
example shows that, as a rule, this is untrue.
Example
a)
82  64  8
b)
(8) 2  64  8
( (8)2  8)
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Slide 7- 11
Simplifying a
For any real number a,
a  a.
2
(The principal square root of a2 is the
absolute value of a.)
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 12
Example
Simplify each expression. Assume that the
variable can represent any real number.
a)
( y  3)
b)
m12
c)
10
2
x
Solution
a)
2
( y  3)  y  3
Since y + 3 might be negative,
absolute-value notation is
necessary.
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Slide 7- 13
Solution continued
b) Note that (m6)2 = m12 and that m6 is
never negative. Thus,
12
m
6
m .
c) Note that (x5)2 = x10 and that x5 might
be negative. Thus,
10
x
5
 x .
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 14
Cube Roots
We often need to know what number was
cubed in order to produce a certain value.
When such a number is found, we say that
we have found the cube root.
For example, 3 is the cube root of 27
because 33 =27.
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Slide 7- 15
Cube Root
The number c is a cube root of a if c 3 = a. In
symbols, we write 3 a to denote the cube root
of a.
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Slide 7- 16
Example
Simplify
3
27 x3 .
Solution
3
27 x3  3x
Since (–3x)(–3x)(–3x) = –27x3
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Slide 7- 17
Odd and Even nth Roots
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, we
do not write it.
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Slide 7- 18
Example Find each of the following.
a) 5 243
b) 5 243
c)
11 11
m
Solution
a) 5 243  3
Since 35 = 243
b) 5 243  3
Since (–3)5 = –243
c)
11 11
m
m
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 19
Example Find each of the following.
a) 4 81
b) 4 81
c)
4
16m
4
Solution
a) 4 81  3
Since 34 = 81
b) 4 81  can't be simplified.
c)
4
4
16m  2m or 2 m
Not a real number
Use absolute-value notation
since m could represent a
negative number
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 7- 20
Simplifying nth Roots
n
a
n
a
n
a
n
Positive
Positive
a
Negative
-a
Positive
Not a real
number
Positive
Negative
Negative
a
Even
a
Odd
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Slide 7- 21