Answers · The mathematics underneath

Why does any number to the power zero equal one?

In shortAny non-zero number to the power zero is one because no copies are multiplied, and an empty product is one, not zero.

Any non-zero number raised to the power zero is 1. The exponent counts how many copies of the base are multiplied together, so $x^0$ means no copies at all. A product with no factors is the empty product, and the empty product is 1 because 1 is the number that leaves a multiplication unchanged. Zero is the exception: $0^0$ is not fixed by that reading and is left undefined in ordinary real arithmetic.

x^0 \times x^3 = x^{0+3} = x^3 \quad\Longrightarrow\quad x^0 = \frac{x^3}{x^3} = 1

Put $x=2$ and count. Three copies of 2 multiply to $2^3=2\times2\times2=8$ . The rule for multiplying powers of the same base says that the exponents add, so $2^0\times2^3=2^{0+3}=2^3=8$ . Therefore $2^0$ is the number which, multiplied by 8, gives 8. Dividing both sides by 8 gives $2^0=8/8=1$ . Nothing in that calculation is special to 2. With base 10, $10^2=100$ , so $10^0\times10^2=10^{0+2}=10^2=100$ , and $10^0=100/100=1$ . With base $-3$ , $(-3)^2=(-3)\times(-3)=9$ , so $(-3)^0=9/9=1$ .

FIG 1

Expression	Exponent arithmetic	Value
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		, so
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The natural guess

The natural guess is 0, because 'no copies' sounds like 'nothing there'. That reading treats the exponent as a quantity being counted rather than as an instruction about multiplication. Multiplying by 0 is not neutral: $0\times8=0$ , not 8. If $x^0$ were 0, then $x^0\times x^3$ would be $0\times x^3=0$ instead of $x^3$ , and the rule $x^a x^b=x^{a+b}$ would fail when $b=3$ . An empty sum is 0 because adding 0 changes nothing. An empty product is 1 because multiplying by 1 changes nothing. The two conventions are different because addition and multiplication have different neutral values.

Counting copies also explains the other exponent law. Three copies of $x$ followed by four copies is seven copies, so $x^3x^4=x^7$ . Four groups of three copies is twelve copies, so $(x^3)^4=x^{12}$ . A zero exponent is the boundary case of that same counting. It is not a new operation with a new rule. It is what remains when the count reaches zero.

The rule that fixes the value

The value is not chosen by taste. Suppose $x$ is non-zero and $x^0$ is some number $c$ . The counting law for exponents, applied with exponents 0 and 3, says $c x^3=x^3$ . Dividing by $x^3$ is allowed exactly because $x$ is non-zero, and it gives $c=1$ . The same argument works with any positive exponent in place of 3. The only thing that could go wrong is division by zero, which is why the base cannot be zero. This is how notation is usually extended: keep the law that already works, and define the new case as whatever keeps it true.

\frac{x^a}{x^a}=1 \qquad\text{and}\qquad \frac{x^a}{x^a}=x^{a-a}=x^0

Read $x^0$ aloud as 'multiply no copies of $x$ '. The result is not 'nothing'; it is the starting value for a product. When you build a product by multiplying factors one at a time, you start at 1 and multiply each factor in. If there are no factors, you never leave the starting value, so the answer is 1. This is why recursive formulas begin a product at 1 and a sum at 0. A reader who starts a product at 0 gets 0 for every empty product and then the whole formula collapses.

Where it stops

The statement stops at zero. With $x=0$ , the cancellation $0^a/0^a$ is $0/0$ for positive $a$ , which is not defined, so the exponent law cannot recover $0^0$ . In ordinary real arithmetic $0^0$ is left undefined. Some counting arguments set $0^0=1$ because an empty product is empty, but that is a convention for that setting, not a value forced by real powers. Limits of the form $f(x)^{g(x)}$ can also approach different values, which is why $0^0$ is called an indeterminate form when it arises that way. For every non-zero base, negative as well as positive, the value is 1.

Worth keeping

An exponent counts copies, so a zero exponent means no copies at all.
The empty product is one because multiplying by one leaves a product unchanged, just as the empty sum is nought because adding nought leaves a sum unchanged.
For any non-zero base, x^0=1 follows from dividing x^a by itself and subtracting the equal exponents.

Where this goes next

This question is one step of Reading the Notation, a course that works through its subject from the beginning. The page that teaches this part of it is Products, Powers and Roots, and it reads the whole way through whether or not you have an account.

What an account changes is the practice: problems standing under every idea, marked with a reason rather than a tick, in a copy of the course that is yours.

Read that page The whole course