Zero and Negative Exponents

Motivational Example

We know any nonzero number divided by itself is equal to 1. Furthermore, when we divide exponentials with the same base we subtract the exponent on the denominator from the exponent on the numerator. Thus if x is not equal to zero,

Definition: If x is a real number, x not equal to zero,

Examples:

(2x)° = 1

2x° = 2(1)

Motivational example:

Consider

Note when you reduce you get 1/x³.

If you use law IV you get the following

.

Thus .

Definition: If x is not equal to zero and n is a positive integer,

Examples:

 Remark: All laws of exponents hold for integer exponents. It matters not whether they are positive, zero, or negative.

Examples:

  When you multiply exponentials with the same base you add the exponents on the factors to obtain the exponent on the answer.

When you raise an exponential to a power you multiply the exponents together. We always write the final answer so all exponents are positive.

When you divide exponentials with the same base you subtract the exponent on the denominator from the exponent on the numerator.

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