Exponential Functions

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Square Roots and Beyond

The number r is a square root square root of x if r2 = x.

• This is usually written

Likewise, r is the cube rootcube root of x if r3 = x.• This is usually written

x rRadicand

Radical

3 x r

Index

Properties of Square Roots

Properties of Square Roots (a, b > 0)

Product Property

Quotient Property

ab a b

a a

b b

18 9 2 3 2

2 2 2

25 525

Simplifying Square Root

The properties of square roots allow us to simplify radical expressions.

A radical expression is in simplest form when:

1.The radicand has no non-trivial perfect-square factor

2.There’s no radical in the denominator

Exercise 3

Write the first 20 terms of the following sequence:

1, 4, 9, 16, …

These numbers are called the Perfect SquaresPerfect Squares.

Simplest Radical Form

Like the number 3/6, is not in its simplest form. Also, the process of simplification for both numbers involves factors.

75

75

325

325

35

Exercise 4

Express each square root in its simplest form.

12 18 24 32 40

48 60 75 83 3300x

Exercise 5

Simplify the expression.

27 98 10 15 8 28

9

64

15

4

11

25

36

49

Conjugates are Magic!

The radical expressions are called conjugatesconjugates.

• The product of two conjugates is always a rational number

Exercise 8

Identify the conjugate of each of the following radical expressions:

1. 7

2. 5- 11

3. 13 9

Rationalizing the Denominator

We can use conjugates to get rid of radicals in the denominator:

The process of multiplying the top and bottom of a radical expression by the conjugate is called rationalizing the rationalizing the denominatordenominator.

5

1 31 3

1 3

5 1 3

1 3 1 3

5 5 3

2

55 3

2

Fancy One

Exercise 10

Simplify the expression.

6

5

17

12

6

7 5

1

9 7

Exercise 11

Simplify the expression.

9

8

19

21

2

4 114

8 3

nth Roots

Finally, r is an nnth rootth root of x if rn = x.• This is usually written

• On a your calculator, cube and nth roots can be found in the MATH Menu

n x r

Index

SLIDE NUMBER #25

Definition of nb1

44

1

77

nn bb 1

4

1

16For any real number b and for any integer, n, n > 1,

Except when b < 0 and n is even.

From the definition, we can say that and that

The expression

is not defined since -16 < 0 and 4 is even. Can you tell why this restriction is needed?

33

1

8)8(

The number is not defined and it is not real.

SLIDE NUMBER #26

Ex. 3: Evaluate 3

2

8

233

2

)8(8 33 88

Method 1:

22 4

Method 2:

3

233

2

)2(8

3

2

1

3

)2(

2)2( 4

Exercise 3

Without a calculator, evaluate the following.

1. 2. 3 216 4 38

1. 2. 3. 4.

Exercise 4

Without a calculator, evaluate the following.

5 24 3 4811 29 7 81

SLIDE NUMBER #29

Ex. 4: Evaluate 2 27 9 243

SLIDE NUMBER #30

Ex. 7: Express in simplest radical form.

6

5

3

2

2

1

2 ba

6

5

6

4

6

3

6

5

3

2

2

1

22 baba

6

1543 )2( ba

6 5432 ba

6 548 ba