Transcript 10.1 Radical Expressions and Graphs

10.1 Radical
Expressions and
Graphs
Objective 1
Find square roots.
Slide 10.1-3
Find square roots.
When squaring a number, multiply the number by itself. To find the
square root of a number, find a number that when multiplied by itself,
results in the given number. The number a is called a square root of the
number a 2.
Square Root
A number b is a square root of a if b2 = a.
Slide 10.1-4
Find square roots. (cont’d)
The positive or principal square root of a number is written with
the symbol
The symbol 
The symbol
.
is used for the negative square root of a number.
, is called a radical sign, always represents the
positive square root (except that
0  0). The number inside the
radical sign is called the radicand, and the entire expression—radical
sign and radicand—is called a radical.
Radical Sign
Radicand
a
Slide 10.1-5
Find square roots. (cont’d)
The statement
is3incorrect. It says, in part, that a positive number equals a
9 
negative number.
Slide 10.1-6
CLASSROOM
EXAMPLE 1
Finding All Square Roots of a Number
Find all square roots of 64.
Solution:
Positive Square Root
64  8
Negative Square Root
 64  8
Slide 10.1-7
CLASSROOM
EXAMPLE 2
Finding Square Roots
Find each square root.
Solution:
169
 13
 225
 15
25
64
25

64
5

8
Slide 10.1-8
CLASSROOM
EXAMPLE 3
Squaring Radical Expressions
Find the square of each radical expression.
Solution:

17
 31
2x  3
2
 17 

2
  31



 17
2
2x  3
2
 31

2
 2x2  3
Slide 10.1-9
Objective 2
Decide whether a given root is rational,
irrational, or not a real number.
Slide 10.1-10
Deciding whether a given root is rational, irrational, or not a
real number.
All numbers with square roots that are rational are called perfect squares.
Perfect Squares
Rational Square Roots
25
25  5
144
144  12
4
9
4 2

9 3
A number that is not a perfect square has a square root that is irrational.
Many square roots of integers are irrational.
Not every number has a real number square root. The square of a real
number can never be negative. Therefore, -36 is not a real number.
Slide 10.1-11
CLASSROOM
EXAMPLE 4
Identifying Types of Square Roots
Tell whether each square root is rational, irrational, or not a real number.
Solution:
irrational
27
36  6
27
2
rational
not a real number
Not all irrational numbers are square roots of integers. For example 
(approx. 3.14159) is a irrational number that is not an square root of an
integer.
Slide 10.1-12
Objective 3
Find cube, fourth, and other roots.
Slide 10.1-13
Find cube, fourth, and other roots.
Finding the square root of a number is the inverse of squaring a number. In a
similar way, there are inverses to finding the cube of a number or to finding
the fourth or greater power of a number.
The nth root of a is written
n
In
n
a.
a ,the number n is the index or order of the radical.
Index
Radical
sign
n
Radicand
a
It can be helpful to complete and keep a list to refer to of third and fourth powers
from 1-10.
Slide 10.1-14
CLASSROOM
EXAMPLE 5
Finding Cube Roots
Find each cube root.
Solution:
3
64
4
3
27
 3
3
512
8
Slide 10.1-15
CLASSROOM
EXAMPLE 6
Finding Other Roots
Find each root.
Solution:
4
81
 4 81
3
 3
4
81
Not a real number.
5
243
3
5
243
 3
Slide 10.1-16
Objective 4
Graph functions defined by radical
expressions.
Slide 10.1- 16
Graph functions defined by radical expressions.
Square Root Function
The domain and range of the square root function are [0, ).
Slide 10.1- 17
Graph functions defined by radical expressions.
Cube Root Function
The domain and range of the cube function are (, ).
Slide 10.1- 18
CLASSROOM
EXAMPLE 7
Graphing Functions Defined with Radicals
Graph the function by creating a table of values. Give the domain and range.
f ( x)  x  2
Solution:
x
f(x)
–2
2  2  0
–1
0
1  2  1
0  2  1.41
2
22  2
Domain: [2, )
Range: [0, )
Slide 10.1- 19
CLASSROOM
EXAMPLE 7
Graphing Functions Defined with Radicals (cont’d)
Graph the function by creating a table of values. Give the domain and range.
f ( x)  3 x  1
Solution:
X
f(x)
0
3
0  1  1
1
3
1 1  0
2
3
4
2 1  1
3
3  1  1.587
3
3
4 1  1.44
Domain: (, )
Range: (, )
Slide 10.1- 20
Objective 5
Find nth roots of nth powers.
Slide 10.1- 21
Find nth roots of nth powers.
a
For any real number a,
2
a 2 | a | .
That is, the principal square root of a2 is the absolute value of a.
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CLASSROOM
EXAMPLE 8
Simplifying Square Roots by Using Absolute Value
Find each square root.
Solution:
15 |15 | 15
2
y 2 | y |
(12) 2 | 12 | 12
2
(

y
)

 |  y || y |
Slide 10.1- 23
Find nth roots of nth powers.
n
a
If n is an even positive integer, then
If n is an odd positive integer, then
n
n
n
a n | a | .
a n  a.
That is, use absolute value when n is even; absolute value is not
necessary when n is odd.
Slide 10.1- 24
CLASSROOM
EXAMPLE 9
Simplifying Higher Roots by Using Absolute Value
Simplify each root.
Solution:
4
(5) 4
| 5 | 5
5
( 5)5
 5 n is odd
 (3)
6
6
  | 3 | 3
 4 m8
 m 2 n is even
3
24
x
18
 ( y ) | y |
6
x
y
6
8
3 6
3
Slide 10.1- 25
Objective 6
Use a calculator to find roots.
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CLASSROOM
EXAMPLE 10
Finding Approximations for Roots
Use a calculator to approximate each radical to three decimal places.
Solution:
17
3
 4.123
9482  21.166
 362
 19.026
4
 9.089
6825
Slide 10.1- 27
CLASSROOM
EXAMPLE 11
Using Roots to Calculate Resonant Frequency
In electronics, the resonant frequency f of a circuit may be found by the
formula f 
1 where f is the cycles per second, L is in henrys, and C is
2 LC
in farads. (Henrys and farads are units of measure in electronics). Find the
resonant frequency f if L = 6  10-5 and
C = 4  10-9.
Solution:
f 
1
2 LC
f 
1
2 (6 10 )(4 10 )
5
9
 324,874
About 325,000 cycles per second.
Slide 10.1- 28