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.

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

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Method 1: Apply the negative exponent rule
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Method 2: Apply the shortcut and write the reciprocal with a positive exponent.
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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)

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Solutions:
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i)

ii)

iii)

iv)

#### Example C

Evaluate the following:

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

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

Denominator:

Numerator and Denominator:

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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.
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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 .

### 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:

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Answers:
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1.

2.

3.

### Explore More

Evaluate each of the following expressions:

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