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

### Watch This

Khan Academy Negative Exponent Intuition

### Guidance

##### 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.

#### Example A

Evaluate the following using the laws of exponents.

**Solution:**

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.

#### Example B

State the following using only positive exponents: (If possible, use shortcuts)

i)

ii)

iii)

iv)

**Solutions:**

i)

ii)

iii)

iv)

#### Example C

Evaluate the following:

**Solution:**

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.

#### Concept Problem Revisited

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 .

### Vocabulary

- Base
- In an algebraic expression, the
is the variable, number, product or quotient, to which the exponent refers. Some examples are: In the expression , ‘2’ is the base. In the expression , ‘’ is the base.*base*

- Exponent
- In an algebraic expression, the
is the number to the upper right of the base that tells how many times to multiply the base times itself. Some examples are:*exponent*

- In the expression , ‘5’ is the exponent. It means to multiply 2 times itself 5 times as shown here: .
- In the expression , ‘4’ is the exponent. It means to multiply times itself 4 times as shown here: .

- Laws of Exponents
- The
are the algebra rules and formulas that tell us the operation to perform on the exponents when dealing with exponential expressions.*laws of exponents*

### Guided Practice

1. Use the laws of exponents to simplify the following:

2. Rewrite the following using only positive exponents.

3. Use the laws of exponents to evaluate the following:

**Answers:**

1.

2.

3.

### Practice

Evaluate each of the following expressions:

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

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 6.4.