How can you use the quotient rules for exponents to understand the meaning of a zero or negative exponent?

### Zero and Negative Exponents

#### The Zero Exponent

Recall that If , then the following would be true:

However, any quantity divided by itself is equal to one. Therefore, which means . This is true in general:

Note that if is not defined.

#### Negative Exponents

Therefore:

This is true in general and creates the following laws for negative exponents:

These laws for negative exponents can be expressed in many ways:

- If a term has a negative exponent, write it as 1 over the term with a positive exponent. For example: and
- If a term has a negative exponent, write the reciprocal with a positive exponent. For example: and
- If the term is a factor in the numerator with a negative exponent, write it in the denominator with a positive exponent. For example: and
- If the term is a factor in the denominator with a negative exponent, write it in the numerator with a positive exponent. For example: and

These ways for understanding negative exponents provide shortcuts for arriving at solutions without doing tedious calculations. The results will be the same.

#### Let's evaluate the following expression using the laws of exponents:

There are two methods that can be used to evaluate the expression.

**Method 1:** Apply the negative exponent rule

**Method 2:** Apply the shortcut and write the reciprocal with a positive exponent.

Applying the shortcut facilitates the process for obtaining the solution.

#### Now, let's rewrite the following expressions with only positive exponents (If possible, use shortcuts):

### Examples

#### Example 1

Earlier, you were asked how to use the quotient rules for exponents to understand the meaning of a zero or negative exponent.

By the quotient rule for exponents, . Since anything divided by itself is equal to 1 (besides 0), . Therefore, as long as .

Also by the quotient rule for exponents, . If you were to expand and reduce the original expression you would have . Therefore, . This generalizes to .

#### Example 2

Evaluate the following:

There are two methods that can be used to evaluate the problem.

**Method 1:** Work with the terms in the problem in exponential form.

Numerator:

Denominator:

Numerator and Denominator:

**Method 2:** Multiply the numerator and the denominator by . This will change all negative exponents to positive exponents. Apply the product rule for exponents and work with the terms in exponential form.

Whichever method is used, the result is the same.

#### Example 3

Use the laws of exponents to simplify the following:

#### Example 4

Rewrite the following using only positive exponents:

#### Example 5

Use the laws of exponents to evaluate the following

### Review

Evaluate each of the following expressions:

Rewrite the following using positive exponents only. Simplify where possible.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.4.