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Exponents and Scientific Notation
P.2
Definition of a Natural Number
Exponent

If b is a real number and n is a natural number,
n
b  b  b b ... b

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
3
2
n
1 1
 2 
3
9
1
 n
b
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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.
3x 4 y11x 4 y
 3x8 y12
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.
4x 
3 3
 64x
9
The Quotient Rule
m
b
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:
3
4
3 2
1

4

4

4
2
4
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Products to Powers
(ab)n = anbn
When a product is raised to a power, raise each
factor to the power.
Text Example
A. -16y4
B. -8y4
Simplify: (-2y)4.
C. 16y4
D. 8y4
Solution
(-2y)4 = (-2)4y4 = 16y4
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Quotients to Powers
n
a
a
   n
b
b

n
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
16
2
   4 
3
81
3
4
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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 the following number in decimal notation:
2.6 X 107
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.
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Write the following number in decimal notation
1.016 X 10-8

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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Text Example
Write each number in scientific notation. a. 4,600,000
b. 0.00023
Solution
a. 4,600,000 = 4.6 x
b. 0.00023 = 2.3 x
10?
10?
Decimal point moves 6 places
Decimal point moves 4 places
4.6 x 106
2.3 x 10-4