1

Introductory Algebra

Exponents &

Scientific Notation

2

Exponents & Scientific Notation

Integer Exponents

Scientific Notation

Power Rules

3

Integral Exponents & Scientific Notation

4

Positive Integral Exponents

5

Negative Integral Exponents

nn

nn

nn

aa

anda

aa

aa

b

b

a

Rules

1

,1

,1

, 1

6

Negative Integral Exponents

If a is any nonzero real number and n is a positive integer, then a-n = 1/an.

(Note that a negative exponent is not the same or related to a negative number.)

1

3

444

9

3

7177

ppp

7

Negative Integral Exponents

If a is any nonzero real number and n is a positive integer, then a-n = 1/an.

(Note that a negative exponent is not the same or related to a negative number.)

3

133

1

3

3

1

1

9

3

1

9

1

3

1

9131

9

3

1322

33

3

3

1

3

8

Negative Integral Exponents

If a is any nonzero real number and n is a positive integer, then a-n = 1/an.

(Note that a negative exponent is not the same or related to a negative number.)

3

133

3

3

9

3 1)2(32

3

1

3

or

9

Negative Integral Exponents

If a is any nonzero real number and n is a positive integer, then a-n = 1/an.

(Note that a negative exponent is not the same or related to a negative number.)

3

1

27

9

3

9

9

33

1

1

3

or

10

Product Rule for Exponents

2 3 2 3 5

2 3 4

. . 2 2 2 2 2 2 2 2 2

( .)

3 5 ?

m n m na a a

e g

note that the bases must be the same

x y xy

11

Product Rule for Exponents

2 3 2 3 5

2 3 4 2 3 4

. . 2 2 2 2 2 2 2 2 2

( .)

3 5 3 5

m n m na a a

e g

note that the bases must be the same

x y xy x x y y

12

Product Rule for Exponents

2 3 2 3 5

2 3 4 2 3 4

2 1 3 4

3 7

. . 2 2 2 2 2 2 2 2 2

( .)

3 5 3 5

15

15

m n m na a a

e g

note that the bases must be the same

x y xy x x y y

x y

x y

13

Zero Exponents

14

Zero Exponents

0333

3

222

21.. ge

15

Changing the Sign of an Exponent

an and a-n are reciprocals, therefore

a-n = (1/an) and (1/a-n) = an

7

1

21

3

11

21

113

21

3

43

4

34

3

aa

a

aaa

a

16

22 3 1

3

2 3 2 3

4 4

2 1 3 4

1 1

2 22. . 2 2

2 2 2 2

( .)

3 3

5 5

3

53

5

mm n

n

aa e g

a

note that the bases must be the same

x y x y

xy x y

x y

x y

17

22 3 1

3

2 3 2 3

4 4

2 1 3 4

1 1

2 22. . 2 2

2 2 2 2

( .)

3 3

5 5

3

53 3

5 5

mm n

n

aa e g

a

note that the bases must be the same

x y x y

xy x y

x y

xx y

y

18

Scientific Notation

Format: a × 10n × used for multiplication

n an integer

Example: 2 × 103 = 2,000

101 a

19

Scientific Notation Examples

5,280 = ?

.14159 = ?

20

Scientific Notation Examples

5,280 = 5.28 × 103

.14159 = 1.4159 × 10-1

21

Converting Standard to Scientific

1. Count the number of places (n) that the decimal point must be moved so that it will follow the first nonzero digit of the number.

2. If the original number was larger than 10, use 10n.

3. If the original number was smaller than 1, use 10-n.

22

Converting Scientific to Standard

1. Determine the number of places to move the decimal point by examining the exponent on the 10.

2. Move to the right for a positive exponent and to the left for a negative exponent.

23

Multiplication Example

(2,000,000) (0.0000000008)

( 2 × 106 ) ( 8 × 10-10 )

= (2)(8)(106)(10-10)

= 16 × 10-4

= 1.6 × 10-3

= 0.0016

Scientific Notation is designed for multiplication and division, not addition and subtraction.

24

Light from the Sun

Distance to the sun ~ 93,000,000 miles

Speed of light ~ 186,000 miles per second

Find how long it takes for the light from the sun to reach the earth.

25

Light from the Sun

Distance to the sun ~ 93,000,000 miles

Speed of light ~ 186,000 miles per second

Find how long it takes for the light from the sun to reach the earth.

57 1086.1000,186103.9000,000,93

26

Light from the Sun

Distance to the sun ~ 93,000,000 miles

Speed of light ~ 186,000 miles per second

Find how long it takes for the light from the sun to reach the earth.

2575

7

57

1051086.1

3.9

1086.1

103.9

1086.1000,186103.9000,000,93

27

Light from the Sun

Distance to the sun ~ 93,000,000 miles

Speed of light ~ 186,000 miles per second

Find how long it takes for the light from the sun to reach the earth.

sec5001051086.1

3.9

1086.1

103.9

1086.1000,186103.9000,000,93

2575

7

57

28

Warm-ups

29

The Power Rules

30

Power of a Power Rule

326

22232

22222222

2222

31

Power of a Power Rule

12

33

x

x

32

Power of a Power Rule

12 22 9 11

3 9 113

1x xx x

x xx

33

Power of a Product

1243443 1622 xxx

34

Power of a Product

43 23x y

35

Power of a Product

4 4 443 2 3 2

12 8

12

8

3 3

1

81

81

x y x y

x y

x

y

36

Power of a Quotient

3

33

3

3

33

273333

3

2733333

xxxxx

x

xxxxxx

37

Power of a Quotient

2

34

3

x

38

Power of a Quotient

9

16

9

4

3

4

3

4

4

3 6232

2

23232

3

xxxx

x

39

Variable Exponents

n

m

n 5

3

2

40

Variable Exponents

mn

n

nm

nnn

nm

nn

m

n

5

5

5

55

5

5

3

2

3

2

3

2

3

22

41

Integral Exponents

42

Warm-ups

43

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