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 Ã— b Ã— b Ã— â€” â€” â€” Ã— b, where there are n factors of b. b is called the base, and n is called the exponent. By definition, b^{0} = 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.
Example1
If x â‰ 0,

x^{5}

x^{6}

x^{7}

x^{8}

x^{9}
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, :
Example2
=
 1/3
 4/9
 4/3
Solution
Canceling the common factor 3 yields , or .
Now, by the definition of a power, .
Hence, the answer is (A).
Now, by the definition of a power, .
Hence, the answer is (A).
Example3
=
Solution
First, factor the top of the fraction:
Next, apply the rule :
Finally, apply the rule :
Hence, the answer is (C).
Hence, the answer is (C).