Exponents and Their Properties
Recall:
if n is a positive
integer.
New Stuff:
When you have a product of two terms with the same base, the
exponents add together.
Rule (The Product Rule)
For any number a and positive integers m and n , a^{ m}
a^{ n} = a ^{m + n}.
 Multiplication of powers leads to addition of the
exponents. When you raise something to a power, you are
really multiplying it by itself several times. Thus, if
you raise a power to a power, you should add the exponent
to itself several times. In other words, multiply the
powers.
Rule (Power Rule)
For any number a and any positive integers m and n , ( a^{
m} )^{ n} .
 Since multiplication leads to addition of exponents, it
makes sense that division would lead to subtraction.
Rule (The Quotient Rule)
For any number a ( a
0) and any positive integers m and n for which m > n ,.
The quotient rule allows us to determine what a 0 exponent
should mean. Suppose that we allow m = n in the rule above. Then
we have (since the top and
bottom are the same). But we also have . Thus, we should define a^{ 0} =
1.
Definition ( 0 as an Exponent)
For any real number a , a
0, a^{ 0} = 1.
 That this is a good way to define 0 as an exponent is
verified by looking at how well this works with the other
rules.
The multiplication rule: If this is a good
definition, then a^{ n} Â· a^{ 0} should be the
same as a^{ n} Â· 1. Indeed, it is, since a^{ n}
Â· a^{ 0} = a^{ n + 0 }= a^{ n} and a^{
n} Â· 1 = a^{ n} .
The power rule: The power rule is really just
an extension of the multiplication rule, so the definition works
with the power rule.
The quotient rule: We made the definition
specifically so it works with the quotient rule, so this works
too.
