# Exponents

Exponents afford a convenient way of expressing long products of the same number. The expression is called a power and it stands for
There are six rules that govern the behavior of exponents:

*b*Ã—*b*Ã—*b*Ã—... 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.

**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 GMAT, 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,

- 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,** :**

Example-2

=

- 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).

Example-3

=

- 2
^{4} - 2
^{3}Ã— 3 - 6
^{2} - 2
^{4}Ã— 3^{2} - 2
^{2}Ã— 3^{4}

Solution

First, factor the top of the fraction:

Next, apply the rule :

Finally, apply the rule :

Hence, the answer is (D).

Hence, the answer is (D).