# What's the arity of the factorial and exponential operations?

I'm having a conflict with the concept of arity, I've read that the factorial is a unary operation and also that the exponentiation is a binary operation but I feel there's something strange, the definition for exponentiation is:

$$b^n = \underbrace{b \times \cdots \times b}_n$$

And the definition for factorial is:

$$n ! = n \times...\times 1$$

So, for both exponentiation ($x^n$) and factorial ($n!$) shouldn't $n$ be the arity? Since we need to perform $n$ multiplications in order to evaluate it? With one exception for $0!=1$ which may be a unary operation.

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When we evaluate the exponential $a^b$ we need two inputs $a$ and $b$. When we evaluate the factorial $n!$ we only need one input $n$.