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Exponents

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:
Example,
Caution,

 

 


Rule 2:

Example,

 

 

 

Rule 3:
Example,

 

 

 

Rule 4:
Example, 

 

 

 

Rule 5: , if a > b.
Example, 

, if b > a.

Example, 


 

 


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.
 

Example-1

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, :

 

Example-2
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).
 

 

Example-3
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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