Fall 2004 1

Working with Powers

• Integer Exponents

• Exponent Rules

• Order of Operations

Fall 2004 2

Examples

• 2 × 2 =

• 2 × 2 × 2 =

• 2 × 2 × 2 × 2 =

• How can we write these in

shorter notation?

22

32

42

Fall 2004 3

Examples

• Multiplication is a shortcut for repeated addition.

• Exponents are a shortcut for repeated multiplication.

3 + 3 + 3 + 3 + 3 = 5 × 3

3 × 3 × 3 × 3 × 3 = 53

Fall 2004 4

Write these numbers using exponential notation:

• 10,000 = 10?

• 27 = 3?

• 32 = 2?

Fall 2004 5

Computer Memory

A byte is capable of storing one letter of the alphabet. For example, the word “math” requires four bytes to store in a computer. Bytes of computer memory are often manufactured in amounts equal to powers of 2.

Fall 2004 6

For Example

1 kilobyte (1K) = 210 =

1 megabyte (1 MB) = 220 =

Fall 2004 7

Integer Exponents

• Base 4• Exponent 3

• 43 is called a power

• 43 = 4 x 4 x 4

• = 64

34

Fall 2004 8

You try these . . .

• 72

• 54

• 210

Fall 2004 9

Operations with Exponents

• Multiply (x3)∙(x4)= (x ∙ x ∙ x) ∙ (x ∙ x ∙ x ∙ x)

= (x ∙ x ∙ x ∙ x ∙ x ∙ x ∙ x) = x7

A5 ∙ A4 =

• Divide: xxx

xxxx

3

4

x

x1

x x

3

4

m

m

Fall 2004 10

Exponent Rules

aaa1an ....#1

n times

Fall 2004 11

Zero as an exponent

• 40

• 90

• 170

Fall 2004 12

Exponent Rules

1a 0#2

Fall 2004 13

a0 = 1

• M0

• (pq)0

• (2x2y)0

Fall 2004 14

Negative Exponents

• 3-1

• 7-1

• 5-2

• 2-4

• (1/2) -1

• (3/4) -2

Fall 2004 15

Exponent Rules

nn

n aa

a 11#3

Fall 2004 16

• M-2

• x-5

• (1/y)-3

nn

aa

1

Fall 2004 17

Combining Exponents

5

32

x

xxxxx

xx

aman=?

Fall 2004 18

Exponent Rules

nmnm aaa #4

Fall 2004 19

• y-2y7

• m6m-6

• 2z-3z5w-6w-2

• x5x4

• b3b-6

• (2x3)(4x-2)

nmnm aaa

Fall 2004 20

Combining Exponents

3

2

5

x

xx

xxxxxx

x

am/an=?

Fall 2004 21

Exponent Rules

nmn

m

aa

a #5

Fall 2004 22

nmn

m

aa

a

2

6

x

x

7

2

y

y

4

4

15

5

m

m

3

5

5

10p

p

Fall 2004 23

Combining Exponents

6

222

32

x

xxxxxx

xxx

x

(am)n=?

Fall 2004 24

Exponent Rules

nmnm aa )(#6

Fall 2004 25

• (w4)5

• (p-3)4

• (5x4)-2

• (-3y-7)3

nmnm aa )(

Fall 2004 26

Combining Exponents3

z

xy

3

33

z

yx

zzz

yyyxxxz

xy

z

xy

z

xy

Fall 2004 27

Exponent Rules

n

nnn

c

ba

c

ab

#7 (distributive rule for exponents)

Fall 2004 28

n

nnn

c

ba

c

ab

432 yx

3

5

2

n

m 2

3

465

c

ba

2475nm

Fall 2004 29

Exponent Rules

• 1.

• 2.

• 3.

aaaan ....

10 a

nn

aa

1

Fall 2004 30

Exponent Rules

• 4.

• 5.

• 6.

• 7.

nmnm aaa

nmn

m

aa

a

nmnm aa )(

n

nnn

c

ba

c

ab