Zero Exponents – Why Anything to the Power of 0 Is 1

It's one of the more surprising rules in mathematics at first glance: any non-zero number raised to the power of 0 equals 1. 7⁰ = 1. 500⁰ = 1. Even a huge, ugly number to the power of 0 is still exactly 1. It doesn't come from a special exception – it falls directly out of the quotient law covered a couple of pages ago, and once you see the proof, it stops feeling like a strange rule and starts feeling inevitable.

Proving x⁰ = 1

Consider xⁿ ÷ xⁿ. Any number divided by itself is 1. But the quotient law also says xⁿ ÷ xⁿ = xⁿ⁻ⁿ = x⁰. Since both must be true at once, x⁰ = 1.

Use the quotient law to show why 6³ ÷ 6³ = 6⁰.

6³ ÷ 6³ = 1 (anything divided by itself is 1). By the quotient law, 6³ ÷ 6³ = 6³⁻³ = 6⁰. So 6⁰ = 1.

Evaluating Zero Exponents

Evaluate 84⁰.

Any non-zero base to the power of 0 is 1: 84⁰ = 1.

Real-Life Application

Algebra: simplifying expressions that reduce to a base with exponent zero.
Polynomials: the constant term of a polynomial is really x⁰ multiplied by a coefficient.
Scientific notation: 10⁰ = 1 is the baseline for the powers-of-10 scale.

Key Takeaways

Any non-zero number raised to the power of 0 equals 1.
This follows directly from the quotient law: xⁿ ÷ xⁿ = x⁰ = 1.
Negative exponents, covered next, extend this same logic below zero.

Practice: Zero Exponents

Evaluate the Zero Exponent