Transcript Slide 1

College Algebra
Sixth Edition
James Stewart  Lothar Redlin

Saleem Watson
P Prerequisites
P.3
Integer Exponents
and Scientific
Notation
Exponents
In this section, we review the rules for
working with exponent notation.
• We also see how exponents can be used to
represent very large and very small numbers.
Exponential Notation
Exponential Notation
A product of identical numbers is usually
written in exponential notation.
• For example, 5 · 5 · 5 is written as 53.
• In general, we have the following definition.
Exponential Notation
If a is any real number and n is a positive
integer, then the nth power of a is:
an = a · a · · · · · a
n factors
• The number a is called the base, and
n is called the exponent.
E.g. 1—Exponential Notation
(a)
 
1 5
2
  21  21  21  21  21  
1
32
(b)  3    3    3    3    3   81
4
(c)  3    3  3  3  3   81
4
Rules for Working with Exponential Notation
We can state several useful
rules for working with exponential
notation.
Rule for Multiplication
To discover the rule for multiplication,
we multiply 54 by 52:
5  5   5  5  5  5  5  5    5  5  5  5  5  5 
4
2
4 factors
2 factors
6 factors
5 5
6
42
• It appears that, to multiply two powers
of the same base, we add their exponents.
Rule for Multiplication
In general, for any real number a and any
positive integers m and n, we have:
a a 
m
n
 a  a  ...  a  a  a  ...  a 
m factors
n factors
 a  a  a  ...  a = a m  n
m  n factors
• Thus, aman = am+n.
Rule for Multiplication
We would like this rule to be true
even when m and n are 0 or negative
integers.
• For instance, we must have:
20 · 23 = 20+3 = 23
• But this can happen only if 20 = 1.
Rule for Multiplication
• Likewise, we want to have:
54 · 5–4 = 54+(–4) = 54–4 = 50 = 1
• This will be true if 5–4 = 1/54.
• These observations lead to the following
definition.
Zero and Negative Exponents
If a ≠ 0 is any real number and n is
a positive integer, then
a0 = 1
and
a–n = 1/an
E.g. 2—Zero and Negative Exponents
(a) 

4 0
7
(b) x
1
1
1 1
 1
x
x
(c)  2 
3

1
 2
3
1
1


8
8
Rules for Working with
Exponents
Laws of Exponents
Familiarity with these rules is essential
for our work with exponents and bases.
• The bases a and b are real numbers.
• The exponents m and n are integers.
Law 3—Proof
If m and n are positive integers, we have:
 a    a  a  ...  a 
m
n
n
m factors
  a  a  ...  a  a  a  ...  a  ...  a  a  ...  a 
m factors
m factors
m factors
n group of factors
 a  a  ...  a  a
mn
mn factors
• The cases for which m ≤ 0 or n ≤ 0 can be proved
using the definition of negative exponents.
Law 4—Proof
If n is a positive integer, we have:
 ab 
n
  ab  ab  ...  ab 
n factors
  a  a  ...  a    b  b  ...  b   a n b n
n factors
n factors
• We have used the Commutative and Associative
Properties repeatedly.
• If n ≤ 0, Law 4 can be proved using the definition
of negative exponents.
E.g. 3—Using Laws of Exponents
(a) x x  x
4
7
4
7
(b) y y
4 7
y
x
4 7
11
y
3
(Law 1)
1
 3
y
(Law 1)
9
c
9 5
4
(c) 5  c  c
c
(Law 2)
E.g. 3—Using Laws of Exponents
 
(d) b
4
5
b
45
b
20
(e)  3 x   3 x  27 x
3
5
3
3
x
x
x
(f )    5 
2
32
2
5
(Law 3)
3
(Law 4)
5
(Law 5)
E.g. 4—Simplifying Expressions with Exponents
Simplify:
3
2
4 3
(a) (2a b )(3ab )
3
x y x
(b)   

y  z 
2
4
E.g. 4—Simplifying
Example (a)
 2a b 3ab 
  2a b  3 a (b ) 
  2a b  27a b 
(Law 3 )
  2  27  a a b b
( Group factors with same base)
 54a b
(Law 1)
3
2
4
3
2
3
2
3
14
3
3
3
6
3
3
4 3
12
2 12
(Law 4 )
E.g. 4—Simplifying
3
4
Example (b)
 
2
4
x
x y x
x y
  3
  
4
y
z
y
z
  

3
8 4
x y x
 3 4
y z
2
3
4
y  1
 x x  3 4
y z

3
7
x y
 4
z
4

(Laws 5 and 4)
(Law 3)
8
(Group factors with
same base)
5
(Laws 1 and 2)
Simplifying Expressions with Negative Exponents
When simplifying an expression, you will
find that many different methods will lead
to the same result.
• You should feel free to use any of the rules
of exponents to arrive at your own method.
E.g. 5—Simplifying Exprns. with Negative Exponents
Eliminate negative exponents and simplify
each expression.
4
6st
(a) 2 2
2s t
 y 
(b)  3 
 3z 
2
E.g. 5—Negative Exponents
Example (a)
We use Law 7, which allows us to move
a number raised to a power from
the numerator to the denominator (or vice
versa) by changing the sign of the exponent:
4
2
6st
6ss
 2 4
2 2
2s t
2t t
3
3s
 6
t
(Law 7)
(Law 1)
E.g. 5—Negative Exponents
Example (b)
We use Law 6, which allows us to change
the sign of the exponent of a fraction by
inverting the fraction.
 y 
 3z 3 


2
 3z 


 y 
9z 6
 2
y
3
2
(Law 6)
(Laws 5 and 4)
Scientific Notation
Scientific Notation
Exponential notation is used by scientists as
a compact way of writing very large numbers
and very small numbers.
For example,
• The nearest star beyond the sun, Proxima Centauri,
is approximately 40,000,000,000,000 km away.
• The mass of a hydrogen atom is about
0.00000000000000000000000166 g.
Scientific Notation
Such numbers are difficult to read and
to write.
So, scientists usually express them
in scientific notation.
Scientific Notation
A positive number x is said to be written
in scientific notation if it is expressed as
follows:
x = a x 10n
where:
• 1 ≤ a < 10.
• n is an integer.
Scientific Notation
For instance, when we state that the distance
to Proxima Centauri is 4 x 1013 km,
the positive exponent 13 indicates that
the decimal point should be moved 13 places
to the right:
4 x 1013 = 40,000,000,000,000
Scientific Notation
When we state that the mass of a hydrogen
atom is 1.66 x 10–24 g, the exponent –24
indicates that the decimal point should be
moved 24 places to the left:
1.66 x 10–24
= 0.00000000000000000000000166
E.g. 6—Changing from Decimal to Scientific Notation
(a)56,920  5.692  10
4
4 places
(b)0.000093  9.3  10
5 places
5
E.g. 7—Changing from Scientific Notation to Decimal
(a) 6.97  10  6,970,000,000
9
9places
(b)4.6271 10
6
 0.0000046271
6places
Scientific Notation in Calculators
Scientific notation is often used on
a calculator to display a very large or
very small number.
• Suppose we use a calculator to square
the number 1,111,111.
Scientific Notation in Calculators
The display panel may show (depending
on the calculator model) the approximation
1.234568 12
or
1.23468
E12
• The final digits indicate the power of 10,
and we interpret the result as 1.234568 x 1012.
E.g. 8—Calculating with Scientific Notation
a ≈ 0.00046
b ≈ 1.697 x 1022
and
c ≈ 2.91 x 10–18
use a calculator to approximate
the quotient ab/c.
If
• We could enter the data using scientific notation,
or we could use laws of exponents as follows.
E.g. 7—Calculating with Scientific Notation

4

4.6  10
1.697  10
ab

c
2.91 10 18
4.6 1.697 

4  22 18

 10
2.91
36
 2.7  10
22

• We state the answer correct to two significant
figures because the least accurate of the given
numbers is stated to two significant figures.