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I have a problem reading a discussion forum post. Namely, in the ASCII text, is 2^3^4 the same as $(2^3)^4$ or $2^{3^4}$?

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There are good answers below, just to note, many programming languages and computer algebra systems assume that power operator ^ or ** is right-associative, i.e. a^b^c = a^(b^c). That kind of rule is also sometimes taught in schools, however, this is not wide-spread, so it is better to clarify with parentheses. Finally, if I had to disambiguate without additional clues, I would default to a^(b^c). – dtldarek May 29 '12 at 14:30
up vote 10 down vote accepted

I believe that usually the intended meaning of a^b^c or $a^{b^c}$ is $a^{(b^c)}$.

The reason is that if someone wants to write $(a^b)^c$, he can use the equivalent expression $a^{bc}$ instead.

In particular, this seems to be quite common in cardinal arithmetic - I think no one will doubt what is meant when someone writes $2^{2^{\aleph_0}}$ even when it's not indicated by brackets: $2^{(2^{\aleph_0})}$.

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I'd say $a^{b^c}$ is unambiguously $a^{(b^c)}$, by convention, but I don't think there is a convention about a^b^c, and I don't think it's safe to make any assumption about the writer's intention. – Gerry Myerson May 30 '12 at 0:16

The reason you have a problem is that the notation is ambiguous. A careful writer will write 2^(3^4) or (2^3)^4, depending on what she means. There is no way of telling what 2^3^4 means, except possibly from context.

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Is the careful writer female? Nice. – Gigili May 29 '12 at 13:52

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