Avon High School
Section: P.2
ACE COLLEGE ALGEBRA II - NOTES
Exponents and Scientific Notation
Mr. Record: Room ALC-129
Day 2
The Product and Quotient Rules for Exponents
The U.S. National Debt as of June 8, 2012 is shown in the box to the right.
Just how big is this number?
How much would each citizen of the U.S. have to pay to eliminate this
debt?
The Product Rule for Exponents
bm  bn  bmn
When multiplying exponential expressions
with the same base, add the exponents. Use
this sum as the exponents of the common
base.
The Quotient Rule for Exponents
bm
 bmn , b  0
bn
When dividing exponential expressions with the same
nonzero base, subtract the exponent in the denominator from
the exponent in the numerator. Use this difference as the
exponents of the common base.
Multiply or divide each expression using the appropriate rule.
Example 1
a.
(4 x3 y 4 )(10 x 2 y8 )
b.
27 x14 y8
3x3 y 5
Zero as an Exponents and Negative Integers as Exponents
The Zero Exponent Rule
If b is any real number other than 0,
b0  1
The Negative-Exponent Rule
If b is any real number other than 0,
and n is a natural number, then
1
bn  n
b
Investigation: Consider the expression below
and use two different methods to deduce that it
is equivalent to 1.
x7
x7
Example 2
a. 5 2
Rewrite each expression with positive exponents.
b. (3) 3
c.
1
4 2
d. 3x 6 y 4
Three Other Rules for Exponents
The Power Rule for Exponents
b 
m n
b
m n
When an exponential expression is raised to
a power, multiply the exponents. Place the
product of the exponents on the base and
remove the parentheses.
The Products-toPowers Rule for
Exponents
n
 ab   a nb n
When a product is
raised to a power, raise
each factor to that
power.
The Quotients-toPowers Rule for
Exponents
n
a
a
   n , b0
b
b
When a quotient is raised
to a power, raise each
part of the fraction to
that power.
n
Simplifying Exponential Expressions
Properties of exponents are used to simplify exponential expressions. An expression is simplified when
1.
2.
3.
4.
Example 3

3
a. 2x y
Simplify.

6 4
b.  6 x y
2
5
 3xy 
3
100 x12 y 2
c.
20 x16 y 4
 5x 
d.  4 
y 
2
Scientific Notation
Sometimes it is much more convenient to write very large (or very small) numbers in
scientific notation.
Scientific Notation Form
A number is written in scientific notation
form when it is expressed as
a  10n
Where the absolute value of a is greater
than or equal to 1 and less than 10
1  a  10  , and n is an integer.
How would you write the U.S.
National Debt from June 8, 2012 in
scientific notation?
Perform the indicated computations, writing each answer in
Example 4
scientific notation for parts a and b.
1.2 106
a.  7.1105  5 107 
b.
3  10 3
Names of Large
Numbers
10 2
hundred
103
thousand
106
million
109
1012
billion
1015
quadrillion
1018
quintillion
10 21
1024
sextillion
1027
octillion
1030
nonillion
10100
10googol
googol
trillion
septillion
googolplex
c. Pell Grants help low-income undergraduate students pay for college. In 2006, the
federal cost of this program was $13 billion and there were 5.1 million grant recipients. How much to the
nearest hundred dollars was the average grant?