# Exponents

*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.

**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.

*If x â‰ 0, *

*A. x ^{5}
B. x^{6}
C. x^{7}
D. x^{8}
E. x^{9}*

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,** :**

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).

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).