Why Does Anything to the Power of Zero Equal 1?

5³ means 5 × 5 × 5. 5² means 5 × 5. 5¹ is just 5. So what's 5⁰, five multiplied by itself zero times? Many people's first instinct is 0. After all, you didn't multiply anything.

The answer is 1. And it's not an arbitrary convention mathematicians agreed on for convenience. It's the only value that keeps the rules of exponents working. Here are three ways to see it, plus the one case mathematicians still argue about.

"Multiplying Nothing" Gives 1, Not 0

When you add nothing, the result is 0. When you multiply nothing, the result is 1.

That sounds strange until you think about what each operation starts from. A running total starts at 0, because adding 0 changes nothing. A running product starts at 1, because multiplying by 1 changes nothing. x⁰ is the product before any multiplications happen, so it's 1.

Explanation 1: Follow the Pattern

Write out the powers of 2 and watch what happens as the exponent goes down:

2⁴ = 16
2³ = 8     (÷ 2)
2² = 4     (÷ 2)
2¹ = 2     (÷ 2)
2⁰ = ?     (÷ 2)

Each step down divides by 2. Continue the pattern: 2 ÷ 2 = 1. So 2⁰ = 1.

Keep going and the pattern also explains negative exponents:

2⁻¹ = 1/2
2⁻² = 1/4

The same pattern works for any non-zero base: dividing 10¹ = 10 by 10 gives 10⁰ = 1. Look at the powers of two in the base-2 logarithm table.

Explanation 2: The Exponent Rules Demand It

When you multiply powers of the same base, you add the exponents:

aᵐ × aⁿ = aᵐ⁺ⁿ

Apply that with n = 0:

aᵐ × a⁰ = aᵐ⁺⁰ = aᵐ

Multiplying aᵐ by a⁰ leaves it unchanged. The only number that does that is 1. So a⁰ = 1.

Or use division. For any non-zero a:

aⁿ / aⁿ = aⁿ⁻ⁿ = a⁰

But any non-zero number divided by itself is 1. So a⁰ = 1. If we defined a⁰ as anything else, these basic laws would break. See the full set on the algebra formulas page.

Explanation 3: Place Value

Our number system already uses zero exponents every day. The number 4,305 means:

4 × 10³ + 3 × 10² + 0 × 10¹ + 5 × 10⁰

The "ones" place is the 10⁰ place. If 10⁰ weren't 1, the last digit of every number would mean something else. The same holds in every base: in binary, the rightmost bit is the 2⁰ = 1 place. See it with the binary to decimal converter.

The Empty Product

Mathematicians call a product with no factors the empty product, and define it as 1. It's the same idea behind:

• 0! = 1: there's exactly one way to arrange zero objects (do nothing)
• A number with no prime factors is 1: the fundamental theorem of arithmetic treats 1 as the empty product of primes
• An empty sum is 0: the additive version of the same idea

This isn't just tidiness. Formulas in combinatorics and algebra work without special cases only because empty products equal 1.

The Exception: What About 0⁰?

Here's where it gets genuinely interesting. Two rules collide:

• "Anything to the power 0 is 1" says 0⁰ = 1
• "0 to any power is 0" says 0⁰ = 0

Calculus sharpens the conflict. As x approaches 0, x⁰ stays at 1, but 0ˣ stays at 0 for positive x. And x^x approaches 1 as x shrinks toward 0 from the positive side. Because different paths give different answers, 0⁰ is called an indeterminate form when it appears as a limit.

An Insider Reference: Knuth Weighs In

In his 1821 textbook Cours d'Analyse, Augustin-Louis Cauchy listed 0⁰ among undefined expressions, alongside 0/0. In the 1830s, mathematician Guglielmo Libri argued for 0⁰ = 1, which sparked an argument that played out in journals.

More than 150 years later, computer scientist Donald Knuth, author of The Art of Computer Programming, made a strong case in his 1992 paper "Two Notes on Notation." He argued that 0⁰ must equal 1 for formulas like the binomial theorem to work:

(x + y)ⁿ = Σ C(n, k) xᵏ yⁿ⁻ᵏ

Set x = 0, and the k = 0 term contains 0⁰. Only 0⁰ = 1 gives the right answer, yⁿ. Knuth wrote that the value of a function 0ˣ at a limit is a separate question from the value of the number 0⁰.

Most programming languages agree: in Python, JavaScript and C, pow(0, 0) returns 1. The IEEE 754 floating-point standard recommends it too.

Two Concepts Worth Knowing

Identity Element

An identity element leaves other values unchanged under an operation. 0 is the identity for addition, and 1 is the identity for multiplication. x⁰ = 1 because an empty product must equal the multiplicative identity.

Indeterminate Form

An indeterminate form, like 0/0 or 0⁰, is an expression whose limit depends on how you approach it. As a plain number, 0⁰ is usually defined as 1, but as a limit it has no single value.

Quick Answer: Why Is Any Number to the Power of 0 Equal to 1?

Because aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰, and any non-zero number divided by itself is 1. The pattern of powers also shows it: each lower exponent divides by the base, so a¹ ÷ a = 1. The case 0⁰ is debated, but it's usually defined as 1 in algebra and programming.

Try Them Yourself

• Base-2 Logarithm Table: powers of two, including 2⁰
• Base-10 Logarithm Table: log₁₀(1) = 0 because 10⁰ = 1
• Algebra Formulas: the laws of exponents
• Binary to Decimal Converter: the 2⁰ place in action
• Scientific Calculator: try 7⁰, 0.5⁰ and 0⁰
• Why Division by Zero Doesn't Work: another place where zero changes the rules

Try 0.1^0.1, then 0.01^0.01, then 0.001^0.001 on a calculator. Watch the results creep toward 1, and you'll see why many mathematicians side with 0⁰ = 1.