laws of exponents. exponential notation base exponent base raised to an exponent

Laws of Exponents

Exponential Notation

BaseExponent

Base raised to an exponent

Goal

To write simplified statements that contain distinct bases, one whole number in the numerator and one in the denominator, and no negative exponents.

Ex:

21 4 3 8 4

2 122 1 2

9 9

46

a b b c

aa b c

Description of Lesson

This power point will address the laws of exponents in the following manner:

1. Explore how the rule works.

2. State the rule.

3. Provide an example with a solution.

4. Provide extra examples (Find me to get solutions)

Exploration

Evaluate the following without a calculator:

34 =

33 =

32 =

31 =

Describe a pattern and find the answer for:

30 =

81

27

9

3

1

÷ 3

÷ 3

÷ 3

÷ 3

Zero Power

Anything to the zero

power is oneCan “a” equal zero?

a0 = 1

No.

You can’t divide by 0.

xxxxxxx

7x

3 4x

Exploration

Simplify: x x3 4

Product of a Power

If you multiply powers having the same base, add the exponents.

m na

Example

Simplify:

32 9x

2 9 03x x y

113x

1

Practice

Simplify the following expressions:

5

3 0 2

3 5 2 4

1)

2) 2 3

3) 9 4

x x

x z x

x y x y

3 3 3 3 3x x x x x

15x

3 5x

Exploration

Simplify: 53x

Power of a Power

To find a power of a power, multiply the exponents.

m na

1 12 9 22 4s s t t

Example

Simplify: 6 32 3 22 4s s t t

13 118s t

2s 2 6s 3 3t 24t

2 4 1 12s 9 2t

Practice

Simplify the following expressions:

42

2 54 3

2 65 4 2

1)

2)

3)

y

a a

x y x x y

5x

2 2 2 2 2z x z x z x z x z x

5 10x z

Exploration

Simplify: 52z x

2 5z

Power of a Product

If a base has a product, raise each factor to the power

m ma b

20y

20y

2x 4 5y 23

Example

Simplify: 52 43 2x xy

7 20288x y

52 5x

2x9 32 5x2 5x 9 32

Practice

Simplify the following expressions:

5

4 52 3

32 4

1)

2) 2 2

3) - 2 3

pqr

ab a

x x yz

Negative Powers

A simplified expression has no negative exponents.

1ma

ma1ma

3b

3

620ba

6 320a b

Example

Simplify: 10 3 44 5a b a

4 5 10 4a

12

2 3128xy

Example

Simplify:2 1

3

12

8

x y

x

532xy

4

4

2x8

3x1y

Practice

Simplify the following expressions:

3

2

5 3

3 8 4

23

1) 8

62)

4

3) 3 y

4) 2

x

x y

x x

a b

x x x x x x x x x xx x x x x x

10 6x

Exploration

Simplify: 10

6xx

4x

Quotient of a Power

To find a quotient of a power, subtract

the bottom exponent from the top if the

bases are the same.

a0

m na

6 2x 26

4 213 x y

Example

Simplify:6

2 3

2

6

x y

x y

4

23xy

1 3y

1

Practice

Simplify the following expressions:6 0

3

6

12

9 3

3

1) 5

122)

4

143)

4

a b

a

x

xy

x y

x y

a a a a a ab b b b b b a a a a a ab b b b b b

Exploration

Simplify:6

a

b

6

6ab

To find a power of a quotient, raise the denominator and numerator to the same power.

Power of a Quotient

m

m

a

b

2y 5 3x 2 6 21

2 15

8

3

y x y

x

Example

Simplify:32 2 7

5

3 2x y

y x

2 6 21

15

8

9

y x y

x

2 3x 7 3y 3223

2 21

15 6

8

9

y

x

23

9

8

9

y

x

Practice

Simplify the following expressions:

2

83

0

4

2

75

4

1)

22)

3)

a

bc

x

y

s f

zr