6.1 Radical Expression Function and Models. Properties of Square Roots Every positive real number has two real-number square roots. The number 0 has just.

6.1 Radical ExpressionFunction and Models

Properties of Square Roots

Every positive real number has two real-number square roots.

The number 0 has just one square root, 0 itself.

Negative numbers do not have real-number square roots.

Square Root

The number c is a square root of a if c2 = a.

1) Find the square roots of 9Answer: 3 and -3

2) Find the square roots of 36Answer: 6 and -6

3) Find the square roots of 1Answer: 1 and -1

4) Find the square roots of -16 Answer: none because no real number

square gives -16

Examples

principal square root (non-negative

square root)

is the radical sign and a is the radicand

Example 1)

5=

0.2=

-4=

10=

a

100

16

2)5(

04.0

Example

Find principal square roots and simplify.

Slide 5

Examples

More examples6) 7)8)

You may have to use calculator if the radicand is not a perfect square

9) f(x) = Find f(0) = f(3)=

f(-5)=

0

16

11

46 x

=0

= 3.32

406 24

436 22

4)5(6 26430 No real answer

No real answer

10) f(x) = 3x + 12

a) Find f(-1)

= 3(-1) + 12 = -3 + 12 = 9

= 3

b) Find f(x+1)

= 3(x+1) + 12

= 3x + 3 + 12 = 3x + 15

For any real number The principal (nonnegative) square root of a2 is the absolute value of a.

Principle Square Root of a2

2)3()12 b

2)10()11

2)1()14 x

4)13 a

2510)15 2 xx

1010

bb 33 b3

2a 2a

1 x

2)5( x 5 x

Radical Expressions and Functions

Cube Root

Find cube roots, simplifying certain expressions, andfind outputs of cube-root functions.

= 2 because 23 =8

5

6

1.0

x

2

0

y2

For any real number :

a) when k is an even natural number. Use absolute value when k is even unless a is nonnegative.

b) when k is an odd natural number greater than 1. Do not use absolute value when k is odd.

Slide 13

Ex:

1) g(x) = 3 x – 3

Find g(-24)

= 3 -24 – 3 = 3 -27

= -3

2) 5 -32 =

= -2

3) 9 (x + 1)9 =

= x + 1

4) 4 81x4 =

= | 3x |

= 3 | x |

5) Find the domain of

Look at the graph using G.C.

Domain: x + 5 ≥ 0

x≥ -5 [-5, ∞)

5x

6) 2x – 5 - 3

Domain is 2x – 5 ≥ 0

2x ≥ 5

x ≥ 5/2

[5/2, ∞)