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Simple question: what does $2^{3^4}$ equal to? I am asking this because some calculators take this as $2^{(3^4)}$ and others as ${(2^3)}^4$.

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    @Martin R: You're right. Thank you! – 2017-01-11
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    Can you give an explicit example or two of calculators that do the evaluation each way? – 2017-01-11
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    I reverted the edit because the notation 2^3^4 to me is quite different from $2^{3^4}$ which almost unambiguously means 2^(3^4),not sure what the OP means. – 2017-01-11

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There is no "correct" answer per se. Because $(2^3)^4 \neq 2^{(3^4)}$, you should really specify which one you mean to avoid confusion. However, we most often decide that exponentiation associates to the right -- that is, $2^{3^4}$ is taken to mean $2^{(3^4)}$ -- because $(2^3)^4$ can also less confusingly be written as $2^{3 \cdot 4} = 2^{12}$.

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    But I learned that $x^{y^z}$ was $x^{(y^z)}$ ... – 2017-01-11
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    @jeff, to avoid confusion, teachers will often say that $x^{y^z}$ *is* $x^{(y^z)}$, just like they will often say that $a + b \cdot c$ *is* $a + (b \cdot c)$. There is no deep truth to these facts. We could more or less just as well have decided that $a + b \cdot c$ means $(a + b) \cdot c$. It is just notation. The important thing is that you communicate in such a way that the reader understands what you mean. To avoid having to write parentheses all the time, we have conventions. The convention $a + b \cdot c = a + (b \cdot c)$ is very strong; that $x^{y^z} = x^{(y^z)}$ is weaker. – 2017-01-12
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    processing this through my head right now.... – 2017-01-13
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If you write $a^{b^c}$ then most people take that as $a^{(b^c)}.$ If you write a^b^c, then almost everyone will stop and ask you where your parens are. Even Maple refuses to do the calculation without parens.

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The answer is $$2^{3^4}$$ $$=2^{81}$$ Which means the order is from top to bottom i.e. $2^{(3^4)}$