Positive, Zero and Negative Exponents

Exponents

1. Positive Exponents:
   Definition: A positive exponent indicates how many times a base number is multiplied by itself.
   Example: 2³ = 2 × 2 × 2 = 8
   Meaning: It represents repeated multiplication.
   
2. Zero Exponents:
   Definition: Any non-zero number raised to the power of zero equals 1.
   Example: 8⁰ = 1, 17⁰ = 1
   Important Note: Some mathematicians define 0⁰ = 1, whereas others leave it undefined.
   
3. Negative Exponents:
   Definition: A negative exponent indicates the reciprocal of the base raised to the positive version of that exponent.
   Example: 2⁻² = 1/2² = 1/4
   (4/3)⁻² = (4/3)² = 16/9
   Meaning: It represents repeated reciprocal of a number.
   

The following diagram shows the pattern when the power of an exponent is reduced by 1. It also gives an idea why a^-b = 1/a^b and why a⁰ = 1. Scroll down the page for the video that will explain this further.

Exponent Worksheets
Practice your math skills with the following exponent worksheets.
Online & Printable Exponent Worksheets

Positive and Negative Exponents
Positive exponents are easy, but what happens when you have zero for the exponent? Or even worse, a negative number?

Pattern for zero and negative exponents
Example:
5³ = 5 • 5 • 5
5² = 5 • 5
5¹ = 5
5⁰ = 1
5^-1 = 1/5
5^-2 = 1/5²
5^-3 = 1/5³

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Zero and negative exponent properties
Zero Exponent Property
Any quantity raised to the zero power equals 1.

Negative Exponent Property
a^-n = the reciprocal of aⁿ
a^-n = 1/aⁿ

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Definitions, covers negative exponents and zero as an exponent

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Negative Exponent Intuition
Intuition on why a^-b = 1/a^b (and why a⁰ = 1)

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Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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