Section P.2

Exponents and Radicals

Definition of Exponent

• An exponent is the power p in an expression ap.

52

• The number 5 is the base.• The number 2 is the exponent. • The exponent is an instruction that tells us

how many times to use the base in a multiplication.

Puzzler

(-5)2 = -52?(-5)(-5) = -(5)(5)

25 = -25

Examples

43

-34

(-2)5

(-3/4)2

=(4)(4)(4) = 64=(-)(3)(3)(3)(3) = -81=(-2)(-2)(-2)(-2)(-2)= -32=(-3/4)(-3/4) = (9/16)

Which of these will be negative?

Multiplication with Exponents by Definition

32 35

= (3)(3) (3)(3)(3)(3)(3)= 37

Note 2+5=7

Property 1 for Exponents

• If a is any real number and r and s are integers, then

To multiply two expressions with the same base, add exponents and use the common base.

sr aa sra

Which one is it?

6x 62x

42 xx

Which one is it?

52 54

32

32 1024

52 23 22

23 22 48

Examples of Property 1

95 925

63 55

By the Definition of Exponents

5

3

xx

2

1x

2x

Notice that 5 – 3 = 2

5

3

xx

5 3x 2x

xxxxxxxx

Examples

3

2

aa

7

4

bb

200

100

cc

1a

a3

1b

3b

100c

Property 2 for Exponents

• If a is any real number and r and s are integers, then

r

s

aa

r sa ( 0)a

To divide like bases subtract the exponents.

Negative Exponents

3

5

xx 2

1x

Notice that 3 – 5 = -2

3

5

xx

3 5x 2x 2

1x

xxxxxxxx

Property 3Definition of Negative Exponents

• If n is a positive integer, then

1 1 nn

naa a

0a

Examples of Negative Exponents

323

12

Notice that: Negative Exponents do not indicate negative numbers.

18

Negative exponents do indicate Reciprocals.

Examples of Negative Exponents

63x6

3x

Notice that exponent does not touch the 3.

Zero as an Exponent2

2

55

2525

1

2

2

55

2 25 05 1

Zero to the Zero?2

2

55

2 25 05 1

2

2

00

2 20 00 1

Zeros are not allowed in the denominator. So 00 is undefined.

STOP

Undefined

Examples08 18 0 14 4

18

1 4 5

Property 5 for Exponents

• If a and b are any real number and r is an integer, then

Distribute the exponent.

rab rr ba

Examples of Property 5

25x 2 25 x 225x

Power to a Power by Definition

(32)3

= ((3)(3))1((3)(3))1((3)(3))1

= 36

Note 3(2)=6

Property 6 for Exponents

• If a is any real number and r and s are integers, then

A power raised to another power is the base raised to the product of the powers.

sra sra

Examples of Property 5 + 6

32 5 4 22 (3 )x y x y 3 6 15 2 8 22 3x y x y6 15 8 28 9x y x y

14 1772x y

))()(98( 21586 yyxx

Examples of Property 6

One base, two exponents… multiply the exponents.

323 63

Definition of nth root of a number

Let a and b be real numbers and let n ≥ 2 be a positive integer. If

a = bn then b is the nth root of a. If n = 2, the root is a square root.If n = 3, the root is a cube root.

Property 2 for Radicals

• The nth root of a product is the product of nth roots

nnn baab

Property 3 for Radicals

• The nth root of a quotient is the quotient of the nth roots

n

nn

ba

ba

4 310 38 2

For a radicand to come out of a radical the exponent must match the index.

a 7 2cb

3 36xd xd 2

5 5a 7 27cb

3 333 xdd

5 425323 zyx 5 zyxzyx

Example 1

50 225

25

Example 2

3448 yx

yyx 34 2

Example 3

3 4540 ba

3 252 baab

Simplified Form for Radical Expressions

A radical expression is in simplified form if1. All possible factors have been removed from

the radical. None of the factors of the radicand can be written in powers greater than or equal to the index.

2. There are no radicals in the denominator.3. The index of the radical is reduced.

Example 6

43

43

23

Example 7

65

65

66

3630

630

Example 10

3 47

3 22

7

3 1

3 1

2

23 3

3

2

27

2273

For something to go inside a radical the exponent must match the index.

3 3a3 1)(a

5 25cb5 2)( cb

72 1)(d 7 14d7 77 )()( dd

Rationalize the denominator.

31

33

93

33

This will always be a

perfect square.

Rationalize the denominator.

57

55

2557

5

57

Often you will not need to

write this step.

Rationalize the denominator.

210

22

4210

2

210

25

5

Simplify numerator first, if possible

1116

114

1111

11114

Simplify first then rationalize.

185

925

235

22

2310

610

Reduce, simplify, rationalize

328

41

41

21

Product

65

21

322

5

32

5

325

33

3215

615

Another product

1024

85

108245

23

23

22

26

Assume that r and t represent nonnegative real numbers.

75 22tr

77

75rt

735rt

Cube roots are a different story.

3

25

3

2222

3

222225

2203

3 3

3

2

45

Must have 3 of a kind

Cube Roots

3

75

3

7777

3

777775

72453

3 3

3

7

495

Must have 3 of a kind

More Cube Roots

3

95

3

33

3

2735

3153

Simplify first

2121518

22

22303

22

153

3

2

4303

Simplify, rationalize

3227

22

21627

24227

2427

8227

Reduce and rationalize

320

2019

33

957

319

357

Cube root

3

52

3

5555

3

125252

5503

P.2 Assignment

• Page 21 • #9 – 42 multiples of 3 (a’s only),• 55 – 63 odd (a’s only)• 72 – 84 Multiples of 3 (a’s only)• 95 – 99 odd (a’s only)