What is 0 to the power of 0?

This is part of a series on common misconceptions.

Is this true or false?

$0^0 = 1$

Why some people say it's true: A base to the power of $0$ is $1$.

Why some people say it's false: An exponent with the base of $0$ is $0$.

The statement $0^0 = 1$ is ambiguous and has been long debated in mathematics.

This is mostly a matter of definition. Mathematicians love to define things. (After all, how else can we talk about mathematics if we don't know the definitions?)

Many sources consider $0^0$ to be an "indeterminate form," or say that $0^0$ is "undefined." On the other hand, other sources/branches of mathematics define $0^0 = 1.$ Note that, certainly, $0^0 \ne 0.$

Some of the arguments for why $0^0$ is indeterminate or undefined are as follows:

Argument 1: We know that $a^0 = 1$ $($for all $a \ne 0),$ but $0^a = 0$ $($for all $a>0).$ This contradiction means $0^0$ should be left undefined.

Argument 2: With respect to limits, if $\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0$, then $\lim_{x \to a} f(x)^{g(x)}$ doesn't necessarily tend to any particular value. For example, $\lim_{x \to 0} e^{-\frac{1}{|x|}} = \lim_{x \to 0} |x| = 0,$ but $\lim_{x \to 0} \left( e^{-\frac{1}{|x|}} \right)^{|x|} = e^{-\frac{1}{|x|} \cdot |x|} = e^{-1}.$ Most of the arguments for why defining $0^0=1$ is useful surround the fact that in some formulas, $0^0=1$ makes the formula true for special cases involving 0.

Example 1: The binomial theorem says that $(x+1)^n \equiv \sum_{k=0}^n \binom{n}{k} x^k$. In order for this to hold for $x = 0$, we need $0^0 = 1$.

Example 2: The power rule in differentiation states that $\frac{d}{dx} x^n = n x^{n-1}$. In order for this to hold for $x = 0$ and $n = 1$, we need $0^0 = 1$.

Example 3: $0^0$ represents the empty product (the number of sets of 0 elements that can be chosen from a set of 0 elements), which by definition is 1. This is also the same reason why anything else raised to the power of 0 is 1.

Rebuttal: $0^0$ has to be $0$, since $0^3 = 0$, $0^2 = 0$, $0^1 = 0$.

Reply: We cannot generalize from that pattern alone. Certainly, $0^{-1}$ does not equal 0, since we cannot divide by 0.

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Rebuttal: $0^0$ has to be $1,$ since formulas like the binomial theorem would not work when $x=0.$

Reply: Mathematics is a subject built upon definitions--there is no "universal truth" of what $0^0$ really equals. If it is convenient for the binomial theorem to assume $0^0=1,$ that is fine. On the other hand, if a mathematician who works with limits chooses to leave $0^0$ as undefined, that is fine too!

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Rebuttal: Why do some problems on Brilliant say that $0^0$ is undefined?

Reply: As explained in this wiki, some sources argue that $0^0$ is undefined. In particular, many algebra courses (prior to the university level) choose to define $0^0$ in this way. Thus, some problem authors--especially in basic algebra problems--may use this definition of $0^0.$

See Also

• Exponents
• Limits of Functions
• List of Common Misconceptions