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5.5
Negative
Exponents and
Scientific Notation
Negative Exponents
Using the quotient rule,
4
x
46
2
x x
6
x
x0
But what does x-2 mean?
x
x x x x
1
1


 2
6
x
x x x x x x x x x
4
Negative Exponents
In order to extend the quotient rule to cases
where the difference of the exponents would
give us a negative number we define negative
exponents as follows.
If a is a real number other than 0, and n is an
integer, then
1
n
a  n
a
Example
Simplify by writing each result using positive
exponents only.
1
1
a. 3  2 
3
9
1
7
b. x  7
x
2
2
c. 2x  4
x
4
Helpful Hint
Don’t forget that since
there are no parentheses,
x is the base for the
exponent –4.
Example
Simplify by write each result using positive
exponents only.
a.  x 3   13
x
2
b.  3
c. (3)
2
1
1
 2 
9
3
1
1

2 
(3)
9
Example
Simplify by writing each of the following expressions
with positive exponents.
a.
b.
1
x 3
x 2
y 4
3
1
x
 x3


1
1
3
x
1
4
2
y
 x  2
1
x
y4
(Note that to convert a power with a
negative exponent to one with a
positive exponent, you simply switch
the power from the numerator to the
denominator, or vice versa, and switch
the exponent to its opposite value.)
Summary of Exponent Rules
If m and n are integers and a and b are real numbers,
then: Product rule for exponents am · an = am+n
Power rule for exponents (am)n = amn
Power of a product (ab)n = an · bn
n
an
a
Power of a quotient    n , c  0
c
c
am
Quotient rule for exponents n  a m  n , a  0
a
Zero exponent a0 = 1, a ≠ 0
1
n
a

, a0
Negative exponent
n
a
Example
Simplify by writing the following expression with
positive exponents.
 3 ab 
 4 7 3 
3 a b 
2
3
2
3 a b
2

3
4
3
 ab

7 3 2
3

2 2
2

a

3 2
b 
2
3   a  b 
4 2
7 2
3 2
34 a 6b 2
 8 14 6
3a b
8 8
8 8
34 a14
a
b
a
b
48 146 26
4 8 8
 8 6 2 6 3 a b
3 a b  4 
3abb
81
3
Scientific Notation
In many fields of science we encounter very
large or very small numbers. Scientific
notation is a convenient shorthand for
expressing these types of numbers.
A positive number is written in scientific
notation if it is written as the product of a
number a, where 1 ≤ a < 10, and an integer
power r of 10: a ×10r.
Scientific Notation
Step 1: Move the decimal point in the original number
so that the new number has a value between 1
and 10
Step 2: Count the number of decimal places the
decimal point is moved in Step 1. If the original
number is 10 or greater, the count is positive. If
the original number is less than 1, the count is
negative.
Step 3: Multiply the new number in Step 1 by 10
raised to an exponent equal to the count found
in Step 2.
Example
Write each of the following in scientific notation.
a.
4700
Move the decimal 3 places to the left, so that the
new number has a value between 1 and 10.
Since we moved the decimal 3 places, and the
original number was > 10, our count is positive 3.
4700 = 4.7  103
b.
0.00047
Move the decimal 4 places to the right, so that the
new number has a value between 1 and 10.
Since we moved the decimal 4 places, and the
original number was < 1, our count is negative 4.
0.00047 = 4.7  10-4
Scientific Notation
In general, to write a scientific notation
number in standard form, move the decimal
point the same number of spaces as the
exponent on 10. If the exponent is positive,
move the decimal point to the right. If the
exponent is negative, move the decimal point to
the left.
Example
Write each of the following in standard notation.
a.
5.2738  103
Since the exponent is a positive 3, we move the
decimal 3 places to the right.
5.2738  103 = 5273.8
b.
6.45  10-5
Since the exponent is a negative 5, we move the
decimal 5 places to the left.
00006.45 10-5 = 0.0000645
Operations with Scientific
Notation
Multiplying and dividing with numbers written in
scientific notation involves using properties of
exponents.
Example
Perform the following operations.
a. (7.3  10-2)(8.1  105)
= (7.3 · 8.1)  (10-2 · 105)
= 59.13  103
= 59,130
1.2 10 4 1.2 10 4
5

0
.
3

10


 0.000003
b.
9
9
4 10
4 10