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```Radicals and Rational
Exponents
9
Section
9.1
Objectives
1 Simplify a perfect-square root.
2 Simplify a perfect-square root expression.
3 Simplify a perfect-cube root.
4 Simplify a perfect nth root.
3
Objectives
5 Find the domain of a square-root function and
a cube-root function.
6 Use a square root to solve an application.
4
1.
Simplify a perfect-square root
5
Simplify a perfect-square root
When solving an equation, we often must find what number
must be squared to obtain a second number a. If such a
number can be found, it is called a square root of a. For
example,
• 0 is a square root of 0, because 02 = 0.
• 4 is a square root of 16, because 42 = 16.
• –4 is a square root of 16, because (–4)2 = 16.
• 7xy is a square root of 49x2y2, because (7xy)2 = 49x2y2.
• –7xy is a square root of 49x2y2, because (–7xy)2 = 49x2y2.
All positive numbers have two real-number square roots:
one that is positive and one that is negative.
6
Example 1
Find the two square roots of 121.
Solution:
The two square roots of 121 are 11 and –11, because
112 = 121
and
(–11)2 = 121
7
Simplify a perfect-square root
To express square roots, we use the symbol
called a
Read as “The positive square root of 121 is 11.”
Read as “The negative square root of 121 is –11.”
8
Simplify a perfect-square root
Comment
The principal square root of a positive number is always
positive. Although 5 and –5 are both square roots of 25,
only 5 is the principal square root.
expression –
represents –5.
9
Simplify a perfect-square root
Square Root of a
If a  0,
is the positive number, the principal square root
of a, whose square is a.
In symbols,
If a = 0,
If a  0,
. The principal square root of 0 is 0.
is not a real number.
10
Simplify a perfect-square root
Because of the previous definition, the square root of any
number squared is that number. For example,
11
Simplify a perfect-square root
Numbers such as 1, 4, 9, 16, 49, and 1,600 are called
integer squares, because each one is the square of an
integer. The square root of every integer square is an
integer.
12
Simplify a perfect-square root
The square roots of many positive integers are not rational
numbers. For example,
is an irrational number. To find
an approximate value of
with a calculator, we enter
these numbers and press these keys.
11
(
)
Using a scientific calculator
(
) 11
Using a graphing calculator
Either way, we will see that
 3.31662479
13
Simplify a perfect-square root
Square roots of negative numbers are not real numbers.
For example,
is not a real number, because no real
number squared equals –9. Square roots of negative
numbers come from a set called imaginary numbers.
14
2.
Simplify a perfect-square root
expression
15
Simplify a perfect-square root expression
If x  0, the positive number x2 has x and –x for its two
square roots. To denote the principal square root of x2, we
must know whether x is positive or negative.
If x  0, we can write
represents the positive square root of x2, which is x.
If x is negative, then –x  0, and we can write
represents the positive square root of x2, which is –x.
16
Simplify a perfect-square root expression
If we do not know whether x is positive or negative, we
must use absolute value symbols to guarantee that
is
positive.
Definition of
If x can be any real number, then
17
Example 3
Simplify each expression. Assume that x can be any real
number.
a.
Write 16x2 as (4x)2.
= |4x |
Because 16x2 = (|4x|)2. Since x could be
negative, absolute value symbols are needed.
= 4|x|
Since 4 is a positive constant in the
product 4x, we write it outside the absolute
value symbols.
18
Example 3
b.
cont’d
Factor x2 + 2x + 1.
Because (x + 1) can be negative,
absolute value symbols are needed.
c.
Because x4 = ( x2)2, and x2  0,
no absolute value symbols are
needed.
19
3.
Simplify a perfect-cube root
20
Simplify a perfect-cube root
The cube root of x is any number whose cube is x. For
example,
4 is a cube root of 64, because 43 = 64.
3x2y is a cube root of 27x6y3, because (3x2y)3 = 27x6y3.
–2y is a cube root of –8y3, because (–2y)3 = –8y3.
21
Simplify a perfect-cube root
Cube Root of a
The cube root of a is denoted as
whose cube is a. In symbols,
and is the number
If a is any real number, then
22
Simplify a perfect-cube root
We note that 64 has two real-number square roots, 8 and
–8. However, 64 has only one real-number cube root, 4,
because 4 is the only real number whose cube is 64.
Since every real number has exactly one real cube root, it
is unnecessary to use absolute value symbols when
simplifying cube roots.
23
Example 4
a.
Because 53 = 5  5  5 = 125
b.
Because
c.
Because (–3x)3 = (–3x)(–3x)(–3x) = –27x3
d.
e.
(0.6xy2)3 = (0.6xy2)(0.6xy2)(0.6xy2)
= 0.216x3y6
24
Simplify a perfect-cube root
Comment
The previous examples suggest that if a can be factored
into three equal factors, any one of those factors is a cube
root of a.
25
4.
Simplify a perfect nth root
26
Simplify a perfect nth root
Just as there are square roots and cube roots, there are
fourth roots, fifth roots, sixth roots, and so on.
is called the index (or order) of the
radical. When the index is 2, the radical is a square root,
and we usually do not write the index.
27
Simplify a perfect nth root
Comment
When n is an even number greater than 1 and x  0,
is
not a real number. For example,
is not a real number,
because no real number raised to the 4th power is –81.
However, when n is odd,
is a real number.
28
Simplify a perfect nth root
When n is an odd natural number greater than 1,
represents an odd root. Since every real number has only
one real nth root when n is odd, we do not need to use
absolute value symbols when finding odd roots. For
example,
because 35 = 243
because (–2x)7 = –128x7
29
Simplify a perfect nth root
When n is an even natural number greater than 1,
represents an even root. In this case, there will be one
positive and one negative real nth root.
For example, the two real sixth roots of 729 are 3 and –3,
because 36 = 729 and (–3)6 = 729.
30
Simplify a perfect nth root
When finding even roots, we use absolute value symbols to
guarantee that the principal nth root is positive.
= |–3| = 3
34 = (–3)4. We also could simplify this as
follows:
= |3x | = 3|x|
(3|x|)6 = 729x6. The absolute value symbols
guarantee that the sixth root is positive.
31
Example 5
a.
b.
c.
because 54 = 625
because (–2)5 = –32
as “the
fourth root of 625.”
as “the
fifth root of –32.”
because
as “the
sixth root of
d.
because
107
=
107
”
as “the
seventh root of 107.”
32
Simplify a perfect nth root
When finding the nth root of an nth power, we can use the
following rules.
Definition of
If n is an odd natural number greater than 1, then
If n is an even natural number, then
33
Simplify a perfect nth root
We summarize the possibilities for
as follows:
Definition for
Assume n is a natural number greater than 1 and x is a real
number.
If x  0, then
is the positive number such that
If x = 0, then
34
Simplify a perfect nth root
If x  0
and n is odd, then
such that
and n is even, then
is the real number
is not a real number.
35
5. Find the domain of a square-root
function and a cube-root function
36
Find the domain of a square-root function and a cube-root function
Since there is one principal square root for every
nonnegative real number x, the equation f(x) =
determines a function, called the square-root function.
37
Example 7
Consider the function f(x) =
.
a. Find the domain. b. Graph the function. c. Find the
range.
Solution:
a. To find the domain, we note that x  0 in the function
because the radicand must be nonnegative. Thus, the
domain is the set of nonnegative real numbers. In
interval notation, the domain is [0, ).
38
Example 7 – Solution
cont’d
b. We can create a table of values and plot points to obtain
the graph shown in Figure 9-1(a). If we use a graphing
calculator, we can choose window settings of [–1, 9] for x
and [–2, 5] for y to see the graph shown in Figure 9-1(b).
(a)
(b)
Figure 9-1
39
Example 7 – Solution
cont’d
c. From either graph, we can conclude that the range of the
function is the set of nonnegative real numbers, which is
the interval [0, ). The graph also confirms that the
domain is the interval [0, ).
40
Find the domain of a square-root function and a cube-root function
The graphs of many functions are translations or reflections
of the square-root function. For example, if k > 0,
• The graph of
translated k units upward.
is the graph of
• The graph of
is the graph of
translated k units downward.
• The graph of
is the graph of
translated k units to the left.
41
Find the domain of a square-root function and a cube-root function
• The graph of
is the graph of
translated k units to the right.
• The graph of
is the graph of
42
Find the domain of a square-root function and a cube-root function
The equation
defines the cube-root function.
From the graph shown in Figure 9-2(a), we can see that the
domain and range of the function
are the set of
real numbers, ℝ, and in interval notation
Figure 9-2(a)
43
Find the domain of a square-root function and a cube-root function
Note that the graph of
passes the vertical line
test. Figures 9-2(b) and 9-2(c) show several translations of
the cube-root function.
(b)
(c)
Figure 9-2
44
6.
Use a square root to solve an
application.
45
Use a square root to solve an application
In statistics, the standard deviation of a data set is a
measure of how tightly the data points are grouped around
the mean (average) of the data set. The smaller the value,
the more tightly the data points are grouped around the
mean.
To see how to compute the standard deviation of a
distribution we consider the distribution 4, 5, 5, 8, 13 and
construct the following table.
46
Use a square root to solve an application
The population standard deviation of the distribution is the
positive square root of the mean of the numbers shown in
column 4 of the table.
47
Use a square root to solve an application
To the nearest hundredth, the standard deviation of the
given distribution is 3.29.
The symbol for the population standard deviation is , the
lowercase Greek letter sigma.
48
Example 10
Which of the following distributions is more tightly grouped
around the mean?
a. 3, 5, 7, 8, 12
b. 1, 4, 6, 11
49
Example 10 – Solution
We compute the standard deviation of each distribution.
a.
50
Example 10 – Solution
cont’d
b.
Since the standard deviation for the first distribution is less
than the standard deviation for the second, the first
distribution is more tightly grouped around the mean.
51
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