In the case where a=3, note that: 3^(3^3) = 3^(3*3*3) = ((3^3)^3)^3 Call the first form x and the last form y. They have different numbers of threes, but we can balance them by using equal numbers of x and y, e.g. x^y = y^x i.e.: (3^(3^3))^(((3^3)^3)^3) = (((3^3)^3)^3)^(3^(3^3)) You can do the same thing for any integer. E.g. in the case of a=4: 4^(4^4) = 4^(4*4*4*4) = (((4^4)^4)^4)^4 Tom Allan Wechsler writes:
Rich politely shielded me from the jeers of the rest of the list in pointing out an "algebraic" identity that applies to 3 as it does to any number, namely:
(a^a)^(a^a) = (a^(a^a))^a
In general, if K and L are two different exponentiation-towers, then (a^K)^L = a^(KL) = (a^L)^K. This inspires two new questions:
(1) Is this "KL law" the only algebraic law that exponentiation-towers obey in general, or are there other such general relations not inferrable from the KL law?
(2) In particular for a=3, are there any equivalences that aren't true for general a?
(Probably a careful reading of Guy and Selfridge [1973], cited by Neil earlier, would answer both questions.)