Download Find Square Roots Find Square Roots of Expressions Containing

Math 35
7.1 "Radical Expressions and Radical Functions"
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Objectives:
* Find square roots and graph the square root function
* Find cube roots and graph the cube root function
* Find nth roots
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Preliminaries:
In this section, we will reverse the squaring process and learn how to …nd square roots of numbers. Then we will
generalize the concept of root and consider cube roots, fourth roots, and so on. Finally, we will discuss a new family of
functions, called radical functions.
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Find Square Roots
De…nition:
"Square Root"
The number b is a square root of the number a if
:
Square Root Notation:
If a is a positive real number,
1:
represents the positive or principal square root of a.
2:
represents the negative square root of a.
3: The principal square root of 0 is 0 :
12 = 1
62 = 36
2
2
112 = 121
2
162 = 256
2 =4
7 = 49
12 = 144
172 = 289
32 = 9
82 = 64
132 = 169
182 = 324
42 = 16
92 = 81
142 = 196
192 = 361
52 = 25
102 = 100
152 = 225
202 = 400
:
Square Roots:
1: If a is a perfect square, then
:
2: If a is a positive number that is not a perfect square, then
3: If a is a negative number, then
:
:
Example 1: (Finding square roots)
Evaluate each square root.
p
a)
225
b)
r
49
4
c)
p
0:36
Find Square Roots of Expressions Containing Variables
De…nition:
p
" x2 "
For any real number x,
:
Example 2: (Simplify square root radical expressions)
Simplify the following expressions.
p
a) 25x2
b)
p
x2 + 2x + 1
Page: 1
c)
p
16x8
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
7.1
Graph the Square Root Function
Example 3: (Graphing)
Graph the following functions and …nd the domain and range of each function.
p
p
a) f (x) = x
b) f (x) = x + 3
y
y
4
x
3
f (x)
4
2
2
1
1
2
4
6
8
10
-1
x
3
-4
x
-2
2
4
6
8
-1
f (x)
x
Find Cube Roots
Cube Root of a
The number b is a cube root of the real number a if
:
All real numbers have one real cube root. A positive number has a positive cube root, a negative number has a negative
cube root, and the cube root of 0 is 0.
Cube Root Notation
The cube root of s is denoted by
De…nition of
p
3
p
3
a: By de…nition,
:
x3
For any real number x,
:
13 = 1
33 = 27
53 = 125
73 = 343
93 = 729
23 = 8
43 = 64
63 = 216
83 = 512
103 = 1000
Example 4: (Simplifying cubic roots)
Simplify
q the following expressions.
1
a) 3 64
b)
p
3
125x3
c)
p
3
27m6 n3
Graph the Cube Root Function
Example 5: (Graphing)
Graph the following functions and …nd the domain and range of each function.
p
p
a) 3 x
b) 3 x + 1
y
y
3
x
f (x)
4
x
f (x)
2
2
1
-8 -6 -4 -2
-1
2 4 6 8
x
-8 -6 -4 -2
2 4 6 8
x
-2
-2
-4
-3
Page: 2
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
7.1
Find nth Roots
Just as there are square roots and cube roots, there are fourth roots, …fth roots, and so on. In general, we have the
following de…nition.
De…nition:
"nth Roots of a" p
The nth root of a is denoted by n a; and
:
The number n is called the index (or order) of the radical.
If n is an even natural number, a must be positive or zero, and b must be positive.
p
Rules for n xn
If x is a real number and n > 1; then
1: If n is an odd natural number,
:
2: If n is an even natural number,
:
Example 6: (Finding nth roots)
Evaluate
q each radical expression, if possible. p
5
1
a) 4 81
b)
105
c)
p
6
64
Example 7: (Simplifying radical expressions)
Simplify each radical expression. Assume that x can be any real number.
q
p
5
8
4
a)
32x5
b)
81 (x + 4)
p
Summary of the De…nitions of n x :
If n is a natural number greater than 1 and x is a real number,
p
1: If x > 0; then n x is the positive number such that
2: If x = 0, then
3: If x < 0
(
and n is odd, then
and n is even, then
:
:
p
n
x is the negative number such that
:
is not a real number.
Page: 3
Notes by Bibiana Lopez