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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, $n^n$?

gxtaillon
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1 Answers1

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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

Deepak
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  • If the $n^n$ is the tetration of $n$, would $n^{n^{n}}$ be the square tetration of $n$? – gxtaillon Jun 03 '14 at 00:20
  • There's no such phrase that I'm aware of (square tetration). The square of $n^n$ is $(n^n)^2 = n^{(2n)}$ by the laws of exponents. What you wrote is a power tower of $n$ with 3 terms, and that actually refers to $n^{(n^n)}$. In a power tower, you work "downward" from the top power. Remember that $(n^n)^n$ is (in general) something else, and this is equal to $n^{(n^2)}$ again by the laws of exponents. I hope that's clearer. – Deepak Jun 03 '14 at 00:28