Zero & Negative Exponents

Look at the Quotient of a Power Rule, , and consider what happens when n = m. When n = m, n - m = 0 and the expression would be x⁰. What does the exponent of zero mean? The following example will explain.

Example Simplify

Using expanded notation:

Using the Quotient of a Power Rule:

Therefore, x⁰ must equal 1.

This works for any value of the base except when it is 0, because an exponent on zero would mean that zero would multiply itself a certain number of times. A product with zero is always zero. This leads to the next power rule.

Zero Exponent Rule : .

Earlier in this unit a negative exponent was addressed in an example. In that example it was stated that: . This relationship will be explained here. Consider the pattern that exists in the following powers.

To get from one result to the one below it, the exponent is decreased by one and the result is divided by 2. Continuing the pattern will produce the values below:

What is the relationship between the following pairs of numbers: 2 and , 4 and , 8 and ? They are reciprocals of each other. Therefore, a negative exponent creates a reciprocal value for all values except zero.

Negative Exponent Rule :

Example 1 Write x^-4 using only positive exponents.

Negative exponents create reciprocals.

Example 2 Simplify and write with positive exponents.

Use other power rules but remember that negative exponents create reciprocals.

Example 3 Simplify .

Example 4 Simplify .

Use the Negative Exponent Rule to rewrite with positive exponents.

To learn how to evaluate a negative exponent, select the following link:

Negative Exponents

Example 1 Simplify .

Example 2 Simplify

For more practice on evaluating negative exponents, select the following link:

Negative Exponents

For more examples with zero and negative exponents, select the following links:

Zero and Negative Exponents #1

Zero and Negative Exponents #2

To solidify your understanding of using zero and negative exponents, visit the following link to Holt, Rinehart and Winston Homework Help Online. It provides examples, video tutorials and interactive practice with answers available. The Practice and Problem Solving section has two parts. The first part offers practice with a complete video explanation for the type of problem with just a click of the video icon. The second part offers practice with the solution for each problem only a click of the light bulb away.

Guided Practice

Zero and Negative Exponents Worksheet

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Zero and Negative Exponents Answer Key

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