Tetration is the next iteration of arithmetic operations after addition, multiplication, and exponentiation. To tetrate a number \(a\) to the order \(n\) is to exponentiate \(a\) with itself \(n\) times:

\[\mbox{ }^n\! a=\underbrace{ {{{a^{a\vphantom{h}}}^{\!\unicode{x22F0}\vphantom{h}}}^{\mbox{}\vphantom{h}} }^{\!a\vphantom{h}} }_{\text{$n$ times}}\]

Note that the base itself is counted as one of the exponents for purposes of tetration.

Just as multiplication is repeated addition and exponentiation is repeated multiplication, so tetration is repeated exponentiation. (The word was formed by use of the Greek prefix tetr, meaning “fourth,” since it is the fourth operation in the sequence beginning with addition.) This operation was popularized by Rudy Rucker in his book Infinity and the Mind, and there remain interesting open questions about it. For instance, it is not known if \(\mbox{ }^n\! q\) is rational for any integer \(n\) and non-integer rational number \(q\).