Describing the exponantiation of a number by itself

Question

When a mathematician says

What is the square of $n$?

It is generally understood that the expected answer would be to multiply $n$ by itself, $n^2$.

Is there a word analogous to square to describe the operation of multiplying $n$ $n$-times, $n^n$?

Answer 1

You can call it "self-exponentiation" or simply n raised to the power of itself to avoid ambiguity. There are a few terms that have been used for the generalisation of this operation, viz. $n^{n^{n^...^n}}$ to k terms, and this has been called "tetration", the "power tower" and they "hyperpower function". When you're just doing it once, it's equivalent to self-exponentiation. You can look up this reference: http://mathworld.wolfram.com/PowerTower.html and this one: http://en.wikipedia.org/wiki/Tetration