1 introductory algebra exponents & scientific notation
TRANSCRIPT
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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