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Properties of Exponents I
A. Product Rule
Notice the following:
Similarly,
In general,
Thus, when we multiply powers , we add exponents .
Example:
Find
(by adding exponents)
B. Quotient Rule
Notice the following:
Similarly,
In general,
Thus, when we divide powers , we subtract exponents .
Example:
Find
(by subtracting exponents )
C. Power Rule
Notice the following:
Similarly,
In general,
Thus, when we take a power of a power , we multiply exponents.
Example:
Find
(by multiplying exponents)
WARNING: Don’t get the product and power rules confused:
D. Zero Power
Notice the following: , but by the quotient rule,
Thus, we see that
Similarly,
The above argument works for any number , except zero, because we would have which is indeterminate.
In general,
E. Negative Exponents
Notice the following:
However by the quotient rule.
Thus,
Similarly,
In general,
Thus, a negative exponent means reciprocal.
Example:
Find
(negative exponent rule)
F. Switch Rule
By using the reciprocal idea for negative exponents, we get the switch rule:
Thus, in a fraction, powers in the top get sent to the
bottom by negative exponents, and
powers in the bottom get sent to the top by negative exponents.
Example:
Rewrite without negative exponents:
(using the switch rule)
G. Multiple Power Rule
By using the power rule idea , we see that if we have a
fraction to a power , we hit every
entry with the power (and multiply exponents).
Thus,
Example:
Find
H. Comments on Rules
1. It is important to memorize these rules, and to not get them confused.
2. Parentheses are important !:
3. If everything “leaves” a numerator or denominator , you leave “1” behind:
For example:
4. With fractions, variables not in a fraction are considered to be in the numerator.
For example:
is the same as
5. When using the quotient rule:
If you use it on a variable, the answer goes in the
numerator. If you don’t use it, the
variable stays put.
For example:
In this example, are put in the numerator, and w stays put.
6. When using the multiple power rule:
It is only correct to use it if the variables are not added or subtracted:
can’t be simplified by this rule .
We will explain how to simplify
later in the course. For the time being,
remember the rule:
DON’T APPLY POWERS ACROSS PLUS OR MINUS SIGNS
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