Exponents and
Scientific Notation
Definition of a Natural Number
Exponent
• If b is a real number and n is a natural
number,
b = b ⋅ b ⋅b ⋅... ⋅b
n
• bn is read “the nth power of b” or “ b to the
nth power.” Thus, the nth power of b is
defined as the product of n factors of b.
Furthermore, b1 = b
The Negative Exponent Rule
• If b is any real number other than 0 and n is
a natural number, then
b −n =
1
bn
1
The Zero Exponent Rule
• If b is any real number other than 0,
b0 = 1.
The Product Rule
b m · b n = b m+n
When multiplying exponential expressions
with the same base, add the exponents. Use
this sum as the exponent of the common
base.
The Power Rule (Powers to Powers)
(bm)n = bm•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.
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The Quotient Rule
bm
m −n
n =b
b
• 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 exponent of the common
base.
Example
• Find the quotient of 43/42
Solution:
43
= 43− 2 = 41 = 4
2
4
Products to Powers
(ab)n = anbn
When a product is raised to a power, raise
each factor to the power.
3
Text Example
Simplify: (-2y)4.
Solution
(-2y)4 = (-2)4y4 = 16y4
Quotients to Powers
n
an
a
=
 
bn
b
• When a quotient is raised to a power, raise
the numerator to that power and divide by
the denominator to that power.
Example
• Simplify by raising the quotient (2/3)4 to the
given power.
Solution:
4
2 4 16
2
  = 4 =
3
81
3
4
Properties of Exponents
1. b− n =
5.
1
bn
bm
m −n
=b
bn
2. b0 = 1 3. b m ⋅ bn = bm + n
6. (ab) = a b
n
n
n
4. (b m ) n = b mn
 an an
7.   = n
 b
b
Scientific Notation
The number 5.5 x 1012 is written in a form called scientific notation. A
number in scientific notation is expressed as a number greater than or equal
to 1 and less than 10 multiplied by some power of 10. It is customary to use
the multiplication symbol, x, rather than a dot in scientific notation.
Text Example
• Write each number in decimal notation:
a. 2.6 X 107
b. 1.016 X 10-8
Solution:
a. 2.6 x 107 can be expressed in decimal notation by moving the
decimal point in 2.6 seven places to the right. We need to add six zeros.
2.6 x 107 = 26,000,000.
b. 1.016 x 10-8 can be expressed in decimal notation by moving the
decimal point in 1.016 eight places to the left. We need to add seven
zeros to the right of the decimal point.
1.016 x 10-8 = 0.00000001016.
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Scientific Notation
To convert from decimal notation to scientific notation, we reverse the procedure.
• Move the decimal point in the given number to obtain a number greater than or
equal to 1 and less than 10.
• The number of places the decimal point moves gives the exponent on 10; the
exponent is positive if the given number is greater than 10 and negative if the
given number is between 0 and 1.
Text Example
Write each number in scientific notation. a. 4,600,000 b. 0.00023
Solution
a. 4,600,000 = 4.6 x 10?
b. 0.00023 = 2.3 x 10?
Decimal point moves 6 places
Decimal point moves 4 places
4.6 x 106
2.3 x 10-4
Exponents and
Scientific Notation
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