Zero plays a role unlike any other number in exponentiation. Place it in the base and the result either collapses or self-destructs. Place it in the exponent and a surprising constant emerges. Let both collide — zero as the base and zero as the exponent simultaneously — and mathematics itself splits into competing answers depending on which branch you ask.
Key Terms
Concepts on This Page
Zero Exponent— a0=1 for a=0, forced by the quotient rule
Raising zero to any positive power gives zero. No amount of repetition changes the outcome:
01=002=003=00100=0
Each multiplication introduces another factor of 0, and a single zero factor is enough to force the entire product to 0. This holds whether the exponent is a natural number, a positive rational, or a positive irrational.
The pattern is absolute: 0n=0 for every n>0. Among all real bases, zero is the only one that produces the same output regardless of the positive exponent applied to it.
Zero as a Base — Negative Exponents
Negative exponents take reciprocals: a−n=an1. Applying this to a base of zero gives:
0−n=0n1=01
Division by zero is undefined. The expression 0−1, 0−2, 0−100 — none of them produce a real number.
The behavior of nearby bases hints at why. As the base a approaches 0 from above, a−2=a21 grows without bound. The values 0.1−2=100, 0.01−2=10,000, 0.001−2=1,000,000 — each step closer to zero sends the result further toward infinity.
At a=0 the growth is not just large — it is undefined. There is no finite number that 0n1 can equal. This is the first point in the exponent framework where zero as a base ceases to function, and it is the reason the negative exponents page requires a=0.
Why 0 cannot take a negative power
The graph of a⁻² = 1/a². At a = 1 it is 1, at a = 0.5 it is already 4, and as a closes in on 0 the curve shoots upward without limit, on both sides. At a = 0 itself the formula asks for 1/0, so there is no point on the graph there: 0 raised to any negative power is undefined.
From here on, every exponent law that involves negative powers carries the condition a ≠ 0.
Zero as an Exponent — Why a0=1
Setting the exponent to zero produces a result that surprises at first glance: a0=1 for every nonzero a. The number 50=1. The number (−3)0=1. The number (0.0001)0=1. The base is irrelevant — the answer is always 1.
The quotient rule provides the algebraic proof. Dividing equal powers gives anan=an−n=a0. But anan=1 for any nonzero a. So a0=1.
A numerical pattern reaches the same conclusion from a different direction. List the descending powers of 3: 33=27, 32=9, 31=3. Each step divides by 3. The next step in the sequence is 3÷3=1, so 30=1. The same descent works for any base.
A third argument frames it in terms of the empty product. The expression an is the product of n copies of a. When n=0, there are no copies — an empty product. By convention, the product of no factors is the multiplicative identity, 1, just as the sum of no terms is the additive identity, 0.
Three distinct arguments, one conclusion. The value a0=1 is not arbitrary — it is the only value consistent with the laws of exponents.
aⁿ is the product of n copies of a; n = 0 gives the empty product, which equals the multiplicative identity
a⁰ = 1
Powers of 5: the table starts at 5⁰ = 1
Reading the table upward, each row is the one below it divided by 5: 125, 25, 5, and then 1, which is why 5⁰ must be 1 and not 0. The zero row is forced by the pattern, exactly as the quotient-rule argument says. Try any base and watch the top row read 1 on the powers table.
Only the base 0 breaks this argument, which is why 0⁰ needs the separate discussion below.
Where Both Directions Collide — 00
The expression 00 sits at the intersection of two patterns that contradict each other.
From the base side: 0n=0 for every positive n. As n decreases toward zero, each value is 0. Following this pattern to n=0 suggests 00=0.
From the exponent side: a0=1 for every nonzero a. As a decreases toward zero, each value is 1. Following this pattern to a=0 suggests 00=1.
Both patterns are valid within their own domains. Neither generalizes cleanly to the point where base and exponent are simultaneously zero. The result depends on which direction you approach from — and that dependence is precisely what makes 00 problematic.
The resolution is not a single universal answer. Different branches of mathematics handle 00 differently, each for good reasons rooted in what that branch needs the expression to do.
Two patterns that meet at 0⁰
Two lines over the same axis. Holding the base at 0 and shrinking a positive exponent t gives 0ᵗ = 0 every time; holding the exponent at 0 and shrinking the base t gives t⁰ = 1 every time. Both patterns head for 0⁰ at t = 0, one at height 0 and one at height 1 (the open circles), so neither can settle its value by itself.
Which value 0⁰ takes therefore depends on the field, as the next two sections show.
00 in Discrete Mathematics
In combinatorics, algebra, and number theory, 00=1 is standard convention — not a tentative choice but a practical necessity baked into foundational formulas.
The empty product argument applies directly. The expression a0 represents the product of zero copies of a, and this product equals 1 regardless of a — including a=0. From this perspective, 00=1 is no more controversial than 50=1.
The binomial theorem requires it. The expansion (x+y)n=∑k=0n(kn)xkyn−k includes terms where x=0 or y=0. At x=0 and k=0, the term (0n)⋅00⋅yn must equal yn for the formula to hold. That forces 00=1.
Power series demand it as well. The exponential series ex=∑n=0∞n!xn evaluated at x=0 begins with the term 0!00. The known value e0=1 requires this term to equal 1, which requires 00=1.
In combinatorics, 00 counts the number of functions from the empty set to the empty set — and there is exactly one such function (the empty function). The count is 1.
00 in Analysis
Calculus treats 00 differently. In analysis, it is classified as an indeterminate form — an expression whose value cannot be determined from its components alone.
The function f(x)=xx approaches 1 as x→0+. Evaluated along this path, 00 appears to equal 1.
But the function g(x)=0x equals 0 for every positive x and approaches 0 as x→0+. Along this path, 00 appears to equal 0.
Other paths yield still other values. The expression f(x,y)=xy can be guided toward 00 along curves that produce any non-negative limit. The destination depends on the route — the hallmark of an indeterminate form.
This places 00 alongside 00, ∞−∞, and 1∞ in the catalog of indeterminate forms that resist a universal value. In this context, assigning 00=1 would mask genuinely different limiting behaviors, so analysis leaves it undefined and evaluates each occurrence through limits on a case-by-case basis.
Context
Value of 0⁰
Why this choice
Where it shows up
Discrete mathematics (algebra, combinatorics, number theory)
1 (by convention)
empty product gives the multiplicative identity; required for foundational formulas to hold
binomial theorem, power series eˣ = Σ xⁿ/n! at x = 0, count of functions ∅ → ∅
Analysis (calculus, real analysis)
indeterminate
value depends on the path of approach; no single limit exists
xˣ → 1 vs 0ˣ → 0 as x → 0⁺; catalog with 0/0, ∞ − ∞, 1∞
How Zero Gets Excluded — The Full Picture
The progression through exponent types tells a clear story about zero as a base: it works at first, then gradually fails as the framework expands.
At the natural exponent stage, zero participates without difficulty. 0n=0 for any n≥1, and no rule is violated.
At the negative exponent stage, zero breaks. The reciprocal 0−n=0n1 demands division by zero. From this point on, the laws require a=0.
At the rational exponent stage, 0m/n still works for positive m/n but fails for negative values — the same division-by-zero problem in a fractional setting.
At the irrational exponent stage, the continuous extension that defines ax for all real x requires a>0. Zero is excluded entirely — it cannot anchor a smooth, complete exponential curve.
The endpoint: exponential functionsf(x)=ax are defined only for positive bases. Zero served as a base for the simplest case and was progressively stripped of eligibility as the demands of the framework grew.
requires a > 0 globally; zero cannot anchor a smooth, complete exponential curve
Summary of Zero in Exponentiation
The four scenarios for zero in an exponential expression each resolve differently. The table below collects them with their canonical expression, the resulting value or status, and the reason behind it — including the discrete-math vs analysis split that makes 00 a context-dependent case.
Scenario
Expression
Value or status
Reason
Zero base, positive exponent
0ⁿ (n > 0)
0
any product with a zero factor is zero
Zero base, negative exponent
0⁻ⁿ
undefined
reduces to 1 / 0ⁿ = 1 / 0 — division by zero
Nonzero base, zero exponent
a⁰ (a ≠ 0)
1
forced by the quotient rule and the empty-product convention
Both zero — discrete math
0⁰
1 (by convention)
required by the binomial theorem, power series, and combinatorial counts
Both zero — analysis
0⁰
indeterminate
limit depends on the path of approach (xˣ → 1 vs 0ˣ → 0)
Frequently Asked Questions
Why do different fields give different answers for 00?
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Because they need different things from it. Combinatorics and algebra set 00=1 so the binomial theorem and power series work — at x=0 the leading term of ex=∑n!xn is 0!00, and e0=1 forces it. Analysis leaves it undefined because the limit depends on the path taken. Neither field is wrong; they answer different questions.Read more →
What is the empty product, and why does it equal 1?
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It is the result of multiplying no factors at all, and it equals 1 because 1 is the multiplicative identity — the value that leaves any product unchanged when included. The empty sum equals 0 for the same reason. This is the cleanest argument for a0=1: it is the product of zero copies of a, whatever a happens to be.Read more →
Why is 0−n undefined rather than zero?
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Because a negative exponent means a reciprocal, and zero has none. 0−2 is 021, which is 01. The behavior nearby shows why no value would serve: 0.1−2=100, 0.01−2=10000, and the values grow without bound as the base shrinks. There is nothing finite for 0−2 to be.Read more →
How many functions are there from the empty set to the empty set?
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Exactly one — the empty function, which assigns nothing to nothing. This matters because the number of functions from a set of size n to a set of size m is mn, so the empty-to-empty case counts 00. Since the answer is one, combinatorics takes 00=1: the convention falls out of the counting rather than being imposed on it.Read more →