How to evaluate powers of powers (i.e. $2^3^4$) in absence of parentheses?

Question

If you look at $2^{3^4}$, what is the expected result? Should it be read as $2^{(3^4)}$ or $(2^3)^4$? Normally I would use parentheses to make the meaning clear, but if none are shown, what would you expect?

(In this case, the formatting gives a hint, because I can either enter $2^{3^4}$ or ${2^3}^4$. If I omit braces in the MathML expression, the output is shown as $2^3^4$. Just suppose all three numbers were displayed in the same size, and the same vertical offset between 2, 3 and 3, 4.)

Answer 1

Barring parentheses, $2^{3^4}$ should definitely be read as, and is equivalent to $\;2^{\left(3^4\right)}$:$$2^{3^4} = 2^{(3^4)} = 2^{81}$$ whereas $${(2^3)}^4 = 2^{3\cdot 4} = 2^{12}$$

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Added:

As pointed out in the comments, it is fairly standard practice that exponents are "right associative" - which is somewhat of a misnomer which should only be taken to mean, evaluate rightmost first: read a^b^c as a^(b^c) and read a^b^c^d as a^[b^(c^d)], and so on. As the example above shows,the exponential operator is not associative. So as you have indicated you typically do, use parentheses, when possible, in your own usage, to avoid any possible confusion.

Answer 2

In the same way that any expressions in brackets inside other brackets are done before the rest of the things in the brackets, I'd say that one works from the top down in such a case.

i.e. because we do $(a*d)$ first in$((a*b)*c)*d$, I'd imagine it'd be the expected thing to do $x^{(y^z)}$