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Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc.
5.1
Exponents and
Scientific Notation
Exponents
Exponents that are natural numbers are
shorthand notation for repeating factors.
34 = 3 • 3 • 3 • 3
3 is the base
4 is the exponent (also called power)
Note by the order of operations that exponents
are calculated before other operations.
The Product Rule
Product Rule for Exponents
If m and n are positive integers and a is a real
number, then
am · an = am+n
Example
Use the product rule to simplify.
32 · 34 = 32+4
z3 · z2 · z5
= 36 = 3 · 3 · 3 · 3 · 3 · 3 = 729
= z3+2+5
= z10
(3y2)(– 4y4) = 3 · y2 (–4) · y4
= 3(–4)(y2 · y4)
= –12y6
Zero Exponent
Zero Exponent
If a does not equal 0, then a0 = 1.
Example:
Simplify each of the following expressions.
50 = 1
(xyz3)0 = x0 · y0 · (z3)0 = 1 · 1 · 1 = 1
–x0 = –(x0) = – 1
Example
Evaluate the following.
50 = 1
(xyz3)0 = x0 · y0 · (z3)0 = 1 · 1 · 1 = 1
–x0 = –(x0) = – 1
The Quotient Rule
Quotient Rule for Exponents
If a is a nonzero real number and m and n are
integers, then
m
a
m n

a
n
a
a0
Example
Use the quotient rule to simplify.
x7
7 4
3

x

x
x4
20 x 6
6 5
1

5x

5
x
or 5 x
5
4x
9a 4 b7   9   a4  b7 
4 1
7 2
3 5

3(
a
)(
b
)

3a
b



3 a
2
2
 
3ab
 b 
Group common
bases together
Negative Exponents
Negative Exponents
If a is a real number other than 0 and n is a positive
integer, then
1
an 
an
Example
Simplify and write with positive exponents only.
3
2
1
1
 2 
3
9
1
1
( 4) 

4
( 4)
256
4
2x
4
2
 4
x
Remember that without parentheses, x is the
base for the exponent –4, not 2x
Example
Simplify and write with positive exponents only.
x 9
1
92
11
x
 x  11
2
x
x
7
2 x 7 y 2
x 71y 2( 5) x 8 y 7
y



5
10 xy
5
5
5x8
3
2
(3 x )( x )
x6
3x 32

x6
3x 1
 6  3x 16  3x 7  3
x
x7
Example
Simplify. Assume that a and t are nonzero integers
and that x is not 0.
x 2a  x 3
2t 1
 x 2a 3
x
x 2t 1
(2t 1)( t 5)
2t 1t 5
t 4


x
t 5

x

x
x
x t 5
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.
Scientific notation
A positive number is written in scientific notation
if it is written as a product of a number a, where
1  a < 10, and an integer power r of 10.
a  10r
Writing a Number in Scientific Notation
1) Move the decimal point in the original number to
the until the new number has a value between 1
and 10.
2) Count the number of decimal places the decimal
point was 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.
3) Write the product of 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
Writing a Scientific Notation Number in
Standard Form
Move the decimal point the same number of
places 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