Download Sect. 2.2 - Mt. San Jacinto College

Sect. 7.1 Radical Expressions &
Radical Functions

Square Roots
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The Principal Square Root
Square Roots of Expressions with Variables
The Square Root Function
Cube Roots
The Cube Root Function
Odd & Even nth Roots
7.1
1
Square Roots
Squaring a Number: 7·7 = 72 = 49
Squaring Negatives: (-7)·(-7) = (-7)2 = 49
The Square Roots
of 49:
49 = 7
49 = -7
7.1
2
Simplifying square roots of numbers

Simplify each: (principal root only)
121  1111  11
25
64

5
8
 
5
8
5
8
 81   9  9
2
0.06
0.0036 
2
7.1
 0.06
3
Finding Function Values
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Evaluate each function for a given value of x
f ( x)  3 x  2
g ( z)   6z  4
for f (1)
for g (3)
f (1)  3(1)  2
g (3)   6(3)  4
f (1)  3  2
g (3)   18  4
f (1)  1  1
try f (9)
g (3)   22  4.69
try g (4)
f (9)  3(9)  2
g (4)   6(4)  4
f (9)  27  2
g (4)    24  4
f (9)  25  5
g (4)    20 no real sol
7.1
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Square Roots of Variable Expressions
25a 
2
(5a)  5a  5 | a |
2
4 x 2  12 x  9 
144 y 32 
(2 x  3) 2  2 x  3
(12 y16 ) 2  12 y16  12 y16
7.1
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The Square Root Function
7.1
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Cube Roots
Cubing a Number:
7·7·7 = 73 = 343
Cubing Negatives: (-7)·(-7)·(-7) = (-7)3 = -343
The Cube Root of a positive number is positive
The Cube Root of a negative number is negative
3
64  (4)  4
3
 64  (4)   4
3
3
3
3
7.1
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Recognizing Perfect Cubes (X)3
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Why? You’ll do homework easier, score higher on tests.
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Memorize some common perfect cubes of integers
1
13
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8
23
27
33
64
43
125
53
216 … 1000
63 … 103
Unlike squares, perfect cubes of negative integers are different:
-216 … -1000
(-6)3 … (-10)3
-1
-8 -27 -64 -125
(-1)3 (-2)3 (-3)3 (-4)3 (-5)3

Flashback: Do you remember how to tell if an integer divides evenly by 3?
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Variables with exponents divisible by 3 are also perfect cubes
x3 = (x)3 y6 = (y2)3 -b15 = (-b5)3
Monomials, too, if all factors are also perfect cubes
a3b15 = (ab5)3 -64x18 = (-4x6)3 125x6y3z51 = (5x2yz17)3
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7.1
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Examples to Simplify
3
1000 
3
103  10
3
1
1
  
3
 3
3
1

27
3
 125a 3 
3
0.216 x 3 y 6 
3
3
( 5 a ) 3   5 a
3
(0.6 xy 2 ) 3  0.6 xy 2
7.1
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The Cube Root Function and its Graph
Here is the basic graph:
x 3 x
0 0
1 1
8 2
1 1
8  2
( x, f ( x))
(0,0)
(1,1)
(8,2)
(1,1)
(8,2)
(1,1)
●
(8,2)
●
●
● (0,0)
(-1,-1)
●
(-8,-2)
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Nth Roots
5
100,000  10  10
4 81
16
5

4

3 4
2
3

2
5
5
6
64  2  2
6
6
 0.00243  (0.3)  0.3
5
7.1
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11
Summary of Definitions
7.1
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Examples to Simplify
5
 32 x 5 
5
(2 x) 5   2 x
4
4
x

81
4
x2
x 
  
3
3
8
x 24 
8
( x 3 )8  x 3
22
( x  5) 44 
8
2
22
(( x  5) 2 ) 22  ( x  5) 2
7.1
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What Next?
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Present Section 7.2
7.1
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