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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.
Slide 10.1-4
Square Root
A number b is a square root of a if b2 = a.
The symbol , is called a radical sign, always represents the
positive square root (except that ). The number inside the
radical sign is called the radicand, and the entire expression—radical
sign and radicand—is called a radical.
The positive or principal square root of a number is written with
the symbol .
0 0
a
Radical SignRadicand
The symbol is used for the negative square root of a number.
Slide 10.1-5
Find square roots. (cont’d)
The statement is incorrect. It says, in part, that a positive number equals a negative number.
9 3
Slide 10.1-6
Find square roots. (cont’d)
Find all square roots of 64.
Solution:
Positive Square Root
Negative Square Root
64 8
64 8
Slide 10.1-7
Finding All Square Roots of a NumberCLASSROOM EXAMPLE 1
Find each square root.
Solution:
169
225
13
15
25
64
25
64 5
8
Slide 10.1-8
Finding Square RootsCLASSROOM EXAMPLE 2
Find the square of each radical expression.
Solution:
17 2
17 17
31 2
31 31
22 3x 222 3x 22 3x
Slide 10.1-9
Squaring Radical ExpressionsCLASSROOM EXAMPLE 3
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
144
4
9
25 5
144 12
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, is not a real number.-36
Slide 10.1-11
Tell whether each square root is rational, irrational, or not a real number.
27 irrational
36 26 rational
27 not a real number
Solution:
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
Identifying Types of Square RootsCLASSROOM EXAMPLE 4
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 a
In the number n is the index or order of the radical.,n a
n a
Radical sign
IndexRadicand
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
Find each cube root.
3 64
3 27
3 512
4
3
8
Slide 10.1-15
Finding Cube Roots
Solution:
CLASSROOM EXAMPLE 5
Find each root.
4 81
4 81
4 81
5 243
5 243
3
3
Not a real number.
3
3
Solution:
Slide 10.1-16
Finding Other RootsCLASSROOM EXAMPLE 6
Objective 4
Graph functions defined by radical expressions.
Slide 10.1- 16
Square Root Function
The domain and range of the square root function are [0, ).
Slide 10.1- 17
Graph functions defined by radical expressions.
The domain and range of the cube function are (, ).
Slide 10.1- 18
Graph functions defined by radical expressions.
Cube Root Function
Graph the function by creating a table of values. Give the domain and range.
( ) 2f x x
x f(x)
–2
–1
0
2
2 2 0 1 2 1
0 2 1.41 2 2 2
Domain: [2, )
Range: [0, )
Slide 10.1- 19
CLASSROOM EXAMPLE 7
Graphing Functions Defined with Radicals
Solution:
3( ) 1f x x
X f(x)
0
1
2
3
4
3 0 1 1 3 1 1 0
3 2 1 1 3 3 1 1.587
Domain: (, )
Range: (, )
3 4 1 1.44
Slide 10.1- 20
CLASSROOM EXAMPLE 7
Graphing Functions Defined with Radicals (cont’d)
Graph the function by creating a table of values. Give the domain and range.
Solution:
Objective 5
Find nth roots of nth powers.
Slide 10.1- 21
For any real number a,
That is, the principal square root of a2 is the absolute value of a.
2a2 | | .a a
Slide 10.1- 22
Find nth roots of nth powers.
Find each square root.
215 1|15 | 5 2( 12) | 12 12|
2y | |y 2( )y | | | |yy
Slide 10.1- 23
CLASSROOM EXAMPLE 8
Simplifying Square Roots by Using Absolute Value
Solution:
If n is an even positive integer, then
If n is an odd positive integer, then
That is, use absolute value when n is even; absolute value is not necessary when n is odd.
n na
| | .n na a
.n na a
Slide 10.1- 24
Find nth roots of nth powers.
Simplify each root.
44 ( 5) | | 55 55 ( 5) i5 s odd n
66 ( 3) | 3 | 3
84 m 2 is even m n
Slide 10.1- 25
CLASSROOM EXAMPLE 9
Simplifying Higher Roots by Using Absolute Value
Solution:
3 24x
186 y3| |y
8x3 66 ( )y
Objective 6
Use a calculator to find roots.
Slide 10.1- 26
Use a calculator to approximate each radical to three decimal places.
17 4.123 362 19.026
3 9482 21.166 4 6825 9.089
Slide 10.1- 27
CLASSROOM EXAMPLE 10
Finding Approximations for Roots
Solution:
In electronics, the resonant frequency f of a circuit may be found by the
formula where f is the cycles per second, L is in henrys, and C is
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.
1
2f
LC
1
2f
LC
5 9
1
2 (6 10 )(4 10 )f
324,874
About 325,000 cycles per second.
Slide 10.1- 28
CLASSROOM EXAMPLE 11
Using Roots to Calculate Resonant Frequency
Solution: