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Sect. 7.1 Radical Expressions & Radical Functions. Square Roots The Principal Square Root Square Roots of Expressions with Variables The Square Root Function Cube Roots The Cube Root Function Odd & Even n th Roots. Square Roots. Squaring a Number: 7 · 7 = 7 2 = 49 - PowerPoint PPT Presentation
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7.1 1
Sect. 7.1 Radical Expressions & Radical Functions
Square Roots 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 2
Square RootsSquaring a Number: 7·7 = 72 = 49Squaring Negatives: (-7)·(-7) = (-7)2 = 49
The Square Roots = 7 of 49: = -749
49
7.1 3
Simplifying square roots of numbers Simplify each: (principal root only)
111111121
85
85
85
6425
9981 2
06.006.00036.0 2
7.1 4
Finding Function Values Evaluate each function for a given value of x
525)9(
227)9(
2)9(3)9(
)9(11)1(
23)1(
2)1(3)1(
)1(23)(
f
f
f
ftryf
f
f
fforxxf
solrealnog
g
g
gtryg
g
g
gforzzg
20)4(
424)4(
4)4(6)4(
)4(69.422)3(
418)3(
4)3(6)3(
)3(46)(
7.1 5
Square Roots of Variable Expressions
161621632
22
22
1212)12(144
32)32(9124
||55)5(25
yyyy
xxxx
aaaa
7.1 6
The Square Root Function
7.1 7
Cube RootsCubing a Number: 7·7·7 = 73 = 343Cubing Negatives: (-7)·(-7)·(-7) = (-7)3 = -343
The Cube Root of a positive number is positiveThe Cube Root of a negative number is negative
4)4(64
4)4(643 33
3 33
7.1 8
Recognizing Perfect Cubes (X)3
Why? You’ll do homework easier, score higher on tests. Memorize some common perfect cubes of integers
1 8 27 64 125 216 … 1000 13 23 33 43 53 63 … 103
Unlike squares, perfect cubes of negative integers are different: -1 -8 -27 -64 -125 -216 … -1000 (-1)3 (-2)3 (-3)3 (-4)3 (-5)3 (-6)3 … (-10)3
Flashback: Do you remember how to tell if an integer divides evenly by 3? 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
7.1 9
Examples to Simplify
23 323 63
3 33 3
3
3
3
3 33
6.0)6.0(216.0
5)5(125
31
31
271
10101000
xyxyyx
aaa
7.1 10
The Cube Root Function and its Graph
Here is the basic graph:
)2,8(28)1,1(11
)2,8(28)1,1(11)0,0(00
))(,(3
xfxxx (8,2) ●(1,1)
●
●(0,0) ●
(-1,-1) ●(-8,-2)
7.1 11
Nth Roots
1010000,100 5 55 2264 6 66
234 4
234
1681 3.0)3.0(00243.0 5 55
7.1 12
Summary of Definitions
7.1 13
Examples to Simplify
222 22222 44
38 838 24
2
4
42
4
8
5 55 5
)5())5(()5(
)(
3381
2)2(32
xxx
xxx
xxx
xxx
7.1 14
What Next? Present Section 7.2
