Zero and Negative Exponents
Simple Explanation
Any nonzero number raised to the power 0 equals 1: a⁰ = 1. A negative exponent means "take the reciprocal of the base raised to the positive exponent": a⁻ⁿ = 1/aⁿ.
Why Do We Need It?
Extending exponents to zero and negative values keeps the exponent rules working consistently for every integer, not just positive whole numbers — essential once you start combining and simplifying expressions with the product and quotient rules.
Formula
Zero and Negative Exponent Rule
a⁰ = 1 and a⁻ⁿ = 1/aⁿ (a ≠ 0)
Any nonzero base raised to the power 0 equals 1; a negative exponent means "take the reciprocal, then raise to the positive exponent".
- a
- — the base (any nonzero real number)
- n
- — a positive integer exponent
When to use it: Whenever an exponent is 0 or negative and you need to rewrite it as a positive-exponent expression (or a plain number).
Worked Example
Evaluate a negative exponent
Evaluate 5⁻².
Why Does This Work?
These rules are defined so the pattern aⁿ⁺¹ = aⁿ × a keeps working as n drops through 1, 0, and negative numbers: dividing by a each time as the exponent decreases by 1 forces a⁰ = 1 and a⁻ⁿ = 1/aⁿ.
Real-Life Example
Shrinking measurements in science
A micrometre is 10⁻⁶ metres — a millionth of a metre.
Negative exponents give scientists a compact way to write extremely small quantities without long strings of zeros.
Practice
Evaluate 7⁰.
EasyWhat is 2⁻³?
MediumCommon mistake
Thinking a negative exponent makes the result negative — it does not. A negative exponent means "reciprocal", not "negative number": 2⁻³ = 1/8, not -8.
Quick Review
- a⁰ = 1, for any nonzero a.
- a⁻ⁿ = 1/aⁿ.
- A negative exponent flips the base to a reciprocal — it does not make the value negative.