Section 1.2 Exponents & Radicals

Objectives:•To review exponent rules•To review radicals•To review rational exponents

Integer Exponents

factorsn an = a · a · · · · · a

The number a is called the base and n is called the exponent.

Ex 1. Simplify

512

4

4

(a)

(b) 3

(c) 3

1 1 1 1 1 12 2 2 2 2 32

3 3 3 3 81

3 3 3 3 81

Zero & Negative Exponents

If a ≠ 0 is any real number and n is a positive integer, then

a0 = 1 and a–n = 1na

Ex 2. Simplify

047

1

3

(a)

(b)

(c) 2

x

1

1

x

3

1 1

82

Laws of ExponentsLaw Example

1)

2)

3)

4)

5)

m na a 2 33 3

m

n

a

a

7

4

5

5

nma 352

nab 6xy

na

b

25

7

Class Work

Simplify1. 2.

3. 4.

5.

4 7x x 4 7y y

9

5

c

c

33 2 42 3s t st

34 3 2

2 3

a b b

ab a

Laws for Negative Exponents

Law Example

6.

na

b

nb

a

23

4

24

3

7.

n

m

a

b

m

n

b

a

2

5

3

4

5

2

4

3

Ex 3 Simplify

a) b)

4

2 2

6

2

st

s t

2

33

y

z

Class Work

6.

7.

32 3

2 3 4

xy z

x y z

22

23

23

a bab c

c

Radicals

The symbol √ means: “the positive square root of.”

Thus,

2means and 0a b b a b

nth Roots

If 24 = 16, then it follows that .

4 16 2

If n is any positive integer, then the principal nth root of a is defined as follows:

If n is even, we must have a ≥ 0 and b ≥ 0.

means nn a b b a

Ex 4. Simplify

a)

b)

c)

d)

4 81

5 32

3 8

4 16

3

2

2

not defined

Properties of nth Roots

Property Example1.

2.

3.

n n nab a b 3 8 27

n

nn

a a

b b 4

16

81

3 38 27 2 3 6

4

4

16 2

381

m n mna a3 729 6 729 3

Property Example

4.

5.

n na a if n is odd 33 5 5

n na a if n is even 44 5 5 5

Ex 5. Simplify

3 4)a x

8 44) 81b x y

) 32 200c

3x x

23x y

4 2 10 2 14 2

Class Work

8.

9.

10.

4 2 124 x y z 3 24xz y

3 32 4a b a b 3 36 2 2 2a b a b

75 48 5 3 4 3 9 3

Rational Exponents

For example,

and,

For example,

1/ n na a

31/ 3x x

/ /( ) or equivalentlym n m m n mnna a a a

23 32 / 3 2x x or x

Ex 6. Simplify

1/ 2

2 / 3

1/ 3

(a) 4

(b) 8

(c) 125

Class Work

11.

12.

13.

14.

1/ 3 7 / 3a a

2 / 5 7 / 5

3 / 5

a a

a

3 / 23 42a b

33 / 4 4

1/ 3 1/ 2

2x y

y x

8 / 3a

9 / 5

6 53 / 5

aa

a3 2 9 2 62 a b

9 4

4 1 2 11 4 388

xy x x y

y

HW #2 p21 1-7odd, 9,12,15,17,23-43odd,

45-65eoo,eoo – every other odd