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

• Show Step-by-step Solutions

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

• Show Step-by-step Solutions

Definitions, covers negative exponents and zero as an exponent

• Show Step-by-step Solutions

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

• Show Step-by-step Solutions

Check out our most popular games!

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.