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Exponents afford a convenient way of expressing long products of the same number. The expression  is called a power and it stands for b × b × b × × b, where there are n factors of bb is called the base, and n is called the exponent. By definition, b0 = 1.
There are six rules that govern the behavior of exponents:


Rule 1:



Rule 2:





Rule 3:




Rule 4:




Rule 5: , if a > b.

, if b > a.




Rule 6:
Example,  Caution, a negative exponent does not make the number negative; it merely indicates that the base should be reciprocated.


For example, .

Problems involving these six rules are common on the test, and they are often listed as hard problems. However, the process of solving these problems is quite mechanical: simply apply the six rules until they can no longer be applied.


If x ≠ 0, 

A.  x5
B.  x6
C.  x7
D.  x8
E.  x9

First, apply the rule  to the expression :


Next, apply the rule :


Finally, apply the rule :


The answer is (C).


Note: Typically, there are many ways of solving these types of problems. For this example, we could have begun with Rule 5, :


Then apply Rule 2, :


Finally, apply the other version of Rule 5, :


Column A Column B

Canceling the common factor 3 in Column A yields , or . Now, by the definition of a power, .


Hence, the columns are equal and the answer is (C).


Column A Column B

First, factor Column A: 


Next, apply the rule :


Finally, apply the rule :


Hence, the columns are equal and the answer is (C).

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