Transcript (1/a -n ) = a n

Exponents &
Scientific Notation
Introductory Algebra
1
Exponents & Scientific Notation
Integer Exponents
Scientific Notation
Power Rules
2
Integral Exponents &
Scientific Notation
3
Positive Integral Exponents
4
Negative Integral Exponents
Rules
n
n
n
1
1
a
b
1
n
1
n
     , a    , a  , and n  a
a
a
b
a
a
5
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.)
7p
4
1
7
 7 4   4
p
p
3
3

1
9
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
3
3
1 1
3

 3
1
1 3 9
9
9
1 9 1 32
1
2 3
1
 3   3  3 3 
3 1 3 1
3
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
3
3
3
1
3 ( 2 )
1
or 1  2  3
3 
9
3
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
1
3
9
9 1
or 1  3 

9
3
27 3
9
Product Rule for Exponents
a a  a
m
n
m n
e.g. 2  2   2  2  2  2  2   2
2
3
2 3
2
5
(note that the bases must be the same.)
 3x y  5xy   ?
2
3
4
10
Product Rule for Exponents
a a  a
m
n
mn
e.g . 2  2   2  2  2  2  2   2
2
3
23
2
5
(note that the bases must be the same.)
 3x y  5xy    3 5  x   x   y  y 
2
3
4
2
3
4
11
Product Rule for Exponents
a a  a
m
n
mn
e.g . 2  2   2  2  2  2  2   2
2
3
23
2
5
(note that the bases must be the same.)
2 3
4
2
3
4

3
x
y
5
xy


3
5
x
x
y
y

           
 15 x 21 y 3 4
 15 x y
3
7
12
Zero Exponents
13
Zero Exponents
3
2
3 3
0
e.g. 1  3  2  2
2
14
Changing the Sign of an Exponent
an and a-n are reciprocals, therefore
a-n = (1/an) and (1/a-n) = an

3

 3a
 1  1  1




3
 3 

4
 21a
 a   21  1
4
a
 3  1  4 a
   3  a 
7
 21  a 
 






15
2  2

22
e.g . 3 
 223  21
2
 2  2  2
am
mn

a
an
(note that the bases must be the same.)
2 3

3
x
 y
 5 xy
4


2
3

3
x
y
    
4
5
x
y
    
3 21 3 4
 x y
5
3 1 1
 x y
5
16
am
mn

a
an
2  2

22
e.g . 3 
 223  21
2
 2  2  2
(note that the bases must be the same.)
 3x y    3  x  y 
 5 xy   5 x   y 
2
3
4
2
3
4
3 21 3 4
 x y
5
3 1 1
3x
 x y 
5
5y
17
Scientific Notation
Format:
a × 10n × used for
multiplication
n an integer
1  a  10
Example: 2 × 103 = 2,000
18
Scientific Notation Examples
5,280 = ?
.14159 = ?
19
Scientific Notation Examples
5,280 = 5.28 × 103
.14159 = 1.4159 × 10-1
20
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.
21
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.
22
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.
23
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.
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.
93,000,000  9.310
7
186,000  1.86 10
5
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.
93,000,000  9.310
7
186,000  1.86 10
5
9.3107
9.3
7 5
2


10

5

10
5
1.86 10 1.86
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.
93,000,000  9.310
7
186,000  1.86 10
5
9.3107
9.3
7 5
2


10

5

10
 500 sec
5
1.86 10 1.86
27
Warm-ups
28
The Power Rules
29
Power of a Power Rule
2   2 2 2 
2 3
2
2
2
 2  22  22  2  26  2 23
30
Power of a Power Rule
x 
x 
2 1
3 3

31
Power of a Power Rule
x 
x 
2 1
3 3
2
x
1
2 9
11
 9 x
 x  11
x
x
32
Power of a Product
 2x    2 x 
3 4
4
3 4
 16 x
12
33
Power of a Product
3x
3
y

2 4

34
Power of a Product
 3x
3
y

2 4
  3
4
x   y 
3 4
1 12 8
 x y
81
12
x

81 y 8
2 4
35
Power of a Quotient
3
x
 x   x  x  x  x
        3 
27
 3   3  3  3  3
3
3
3
3
 x
 3  3 3 3 27
        3
x x x x
3
 x
36
Power of a Quotient
2
 3 
 3  
 4x 
37
Power of a Quotient
 3 
 3 
 4x 
2
 4x
  
 3
3
2


 4x
 
2
3

   4 x 
3 2
2
9
3 2
16 x

9
38
6
Variable Exponents
2 
 m 
3 
n
5n
39
Variable Exponents
5n
 
 
2 
2
 m  
m
3
3
 
n
n
5n
5n
n5 n
5n2
2
2
 m5n  5mn
3
3
40
Integral Exponents
41
Warm-ups
42
Next Slide Show
43