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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, 

  1. x5
  2. x6
  3. x7
  4. x8
  5. x9
Solution
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

=

  1. 1/3
  2. 4/9
  3. 4/3
Solution
Canceling the common factor 3 yields , or .
Now, by the definition of a power, .
Hence,  the answer is (A).
 
 
Example-3

=

Solution
First, factor the top of the fraction:
Next, apply the rule :
Finally, apply the rule
Hence, the answer is (C).
 





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