Download Day 6.2008 exponentials

Physics Day 5
Objectives
• SWBAT do exponential math
• Understand factors of 10
Agenda
Do Now
Notes
Worksheet
HW2 due tonight HW3 Thursday
Kiwi thinking about those
Physics experiments with
apples
Day 5
Exponential Math
Exponential Expression
a
n
• An exponential expression is:
where
is called the base and n is
called the exponent
• An exponent applies only to what it is
immediately adjacent to (what it touches)
• Example:
2
3x Exponent applies only to x, not to 3
4
 m Exponent applies only to m, not to negative
3
2x  Exponent applies to (2x)
a
Meaning of Exponent
• The meaning of an exponent depends on
the type of number it is
• An exponent that is a natural number
(1, 2, 3,…) tells how many times to
multiply the base by itself
2
3x  3  x  x
• Examples:
 m  1 m  m  m  m
3
3
2x   2x 2x 2x  8x
4
In the next section we will learn the meaning of any integer exponent
Rules of Exponents
• Product Rule: When two exponential
expressions with the same base are
multiplied, the result is an exponential
expression with the same base having an
exponent equal to the sum of the two
exponents
m
n
m n
a a  a
• Examples:
4 2
3 3  3  3
11
7
4
7 4
x x  x  x
4
2
6
Rules of Exponents
• Power of a Power Rule: When an
exponential expression is raised to a
power, the result is an exponential
expression with the same base having an
exponent equal to the product of the two
exponents
m n
mn
• Examples:
a 
3 
x 
a
4 2
 3
7 4
 x
42
74
 3
28
 x
8
Rules of Exponents
• Power of a Product Rule: When a
product is raised to a power, the result is
the product of each factor raised to the
n
power
n n
• Examples:
ab 
a b
3x 
 3 x  9x
2 y 
 2 y  16 y
2
4
2
4
2
4
2
4
Rules of Exponents
• Power of a Quotient Rule: When a
quotient is raised to a power, the result is
the quotient of the numerator to the power
and the denominator to the power
n
• Example:
a
a
   n
b
b
2
2
3
3
   2 
x
x
n
9
2
x
Rules of Exponents
• Don’t Make Up Your Own Rules
• Many people try to make these rules:
a  b   a  b
n
n
n
a  b   a  b
n
• Proof:
n
n
NOT TRUE! !!!!
NOT TRUE! !!!!
3  2  3  2
2
2
2
3  2  3  2
2
2
2
Using Combinations of Rules to
Simplify Expression with Exponents
• Examples:
52m p
  5  2 m p 5 16m p  80m p
 5x y    5  x y  125x y
2 x y   3x y   8x y  9x y  72x y
2 x y   8x y  8 x
9y
 3x y  9 x y
2
2
3 4
2
3 3
3 3
2
5 2
8
12
3
2
3 3
2
4
6
3 2
9
4
10
12
6
9
6
6
8
9
4
2
6
8
12
9
10 15
Integer Exponents
• Thus far we have discussed the meaning
of an exponent when it is a natural
(counting) number: 1, 2, 3, …
• An exponent of this type tells us how many
times to multiply the base by itself
• Next we will learn the meaning of zero and
negative integer exponents
0
• Examples:
5
2
3
Integer Exponents
• Before giving the definition of zero and
negative integer exponents, consider the
4
pattern: 2 4  16
3  81
3
2 8
33  27
32  9
22  4
1
1
3 3
2 2
0
0
1
3

2 1
1
1
1
1
1
1  
 
1
1

 
 
3

2 
2
22 
2  2
1 1
 
4 2
2
3 
3  3 2
1 1
 
9 3
Definition of Integer Exponents
• The patterns on the previous slide suggest
the following definitions: a 0  1
a
n
1
 
a
n
• These definitions work for any base,
that is not zero:
3
5  1
0
1
1
2   
8
2
3
a,
Quotient Rule for Exponential
Expressions
• When exponential expressions with the same base are divided, the
result is an exponential expression with the same base and an
exponent equal to the numerator exponent minus the denominator
exponent
am
mn

a
an
Examples:
54
47
3
5

5

57
.
x12
12 4
8

x

x
x4
“Slide Rule” for Exponential
Expressions
• When both the numerator and denominator of
a fraction are factored then any factor may
slide from the top to bottom, or vice versa, by
changing the sign on the exponent
Example: Use rule to slide all factors to other
part of the fraction:
a mb  n
cr d s
 m n
r s
c d
a b
• This rule applies to all types of exponents
• Often used to make all exponents positive
Simplify the Expression:
(Show answer with positive exponents)
 
2
16
8y y
8 y 6 y 2
8  21
8 y 8
 1 4 1  1 3  3 8  11
1 4 1
y
2 y y
2 y y
y y
2 y
3
Scientific Notation
• A number is written in scientific notation when it
is in the form:
a 10 , where 1  a  10 and n is an integer
n
Examples:
3.2  105
1.5342 10 9
20
 6.98 10
• Note: When in scientific notation, a single nonzero digit precedes the decimal point
Converting from Normal Decimal
Notation to Scientific
Notation
n
a 10
• Given a decimal number:
– Move the decimal to the right of the first non-zero digit
to get the “a”
– Count the number of places the decimal was moved
• If it was moved to the right “n” places, use “-n” as the
exponent on 10
• If it was moved to the left “n” places, use “n” as the exponent
on 10
• Examples:
320,000
Move decimal 5 places left
3.2  105
0.0000000015342 Move decimal 9 places right 1.5342 10 9
.
 698,000,000,000,000,000,000 Move decimal 20 places left
 6.98 10 20
Converting from Scientific
Notation to Decimal Notation
• Given a number in scientific notation: a 10
n
– Move the decimal in “a” to the right “n” places,
if “n” is positive
– Move the decimal in “a” to the left “n” places,
if “n” is negative
• Examples:
3.2  105
Move decimal 5 places right
1.5342 10
 6.98 10
20
9
Move decimal 9 places left
320,000
0.0000000015342
.
Move decimal 20 places right
 698,000,000,000,000,000,000
Applications of Scientific
Notation
• Scientific notation is often used in situations where the
numbers involved are extremely large or extremely small
• In doing calculations involving multiplication and/or
division of numbers in scientific notation it is best to use
commutative and associative properties to rearrange and
regroup the factors so as to group the “a” factors and
powers of 10 separately and to use rules of exponents to
end up with an answer in scientific notation
• It is also common to round the answer to the least
number of decimals seen in any individual number
Example of Calculations
Involving Scientific Notation
• Perform the following calculations, round
the answer to the appropriate number of
places and in scientific notation
3.2 10  6.98 10   3.2 6.98  10 10  
1.53
10 
1.5310 
14.59869281 10 10 10   14.5986928110 
5
5
20
20
9
9
5
20
9
34
What do we need to do to put this in scientific notation?
1.4598692811035  1.5  1035
Review
a a  a
m
n
a 
m n
n
a
mn
a
a
   n
b
b
n
n
m n
a
n
1
1
   n
a
a
m
a
mn
a
n
a