# 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 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 â‰  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 :

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, .