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

ftry

f

f

f

fforxxf

solrealnog

g

g

gtry

g

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(64

3 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

3

1

3

1

27

1

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

1010000,1005 55 2264

6 66

2

34 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