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I know this one is trivial, but I was wondering: if I have something like $$a^{b^c}$$ then i know that it should be read as $$a^{\left(b^c\right)}$$ if no other parenthesis is present.

Question: if $a=k^2$ for some $k\in\mathbb{N}$, $b=h^2$ for some $h\in\mathbb{N}$, and $c=j^2$ for some $j\in\mathbb{N}$, does the following relation stand true? $$a^{b^c}=k^{k^{h^{h^{j^j}}}}=k^{k\cdot h\cdot h\cdot j\cdot j}$$

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No, if $a = c = 4 = 2^2$ and $b = 1 = 1^2$, then the left hand side is $4^{1^4} = 4^1 = 4$, and the right hand side is $2^{8} \not= 4$.

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Right, this is a degenerate case I didn't consider. I meant positive numbers greater than one. – user138364 Apr 16 '14 at 11:59
Take $b = 4$ as well then. Then the right hand side is 4294967296 while the left hand side is 13407807929942597099574024998205846127479365820592393377723561443721764030073546‌​976801874298166903427690031858186486050853753882811946569946433649006084096. – fuglede Apr 16 '14 at 12:12

$$a^{b^{c}}=\left(k^{2}\right)^{\left(h^{2}\right)^{(j^{2})}}=\left(k^{2}\right)^{h^{2j^{2}}}=k^{2h^{2j^{2}}}$$ Based on: $$\left(r^{s}\right)^{t}=r^{st}$$

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The reason that we unambiguously read $$ a^{b^c} = a^{(b^c)} $$ is that if we meant the parentheses to go the other way, as $(a^b)^c$, then we'd just write $a^{bc}$ instead.

It should be clear that $a^{(b^c)} \neq a^{bc}$, since $b^c \neq bc$. Similarly, $$ k^{k^{h^{h^{j^j}}}} \neq k^{khhjj}, \phantom{NNNNN}(1) $$ since the exponents are visibly different: $$ k^{h^{h^{j^j}}} \neq khhjj. $$ Please note that the two sides of (1) follow opposite association rules: $$ k^{k^{h^{h^{j^j}}}} = k^{(k^{(h^{(h^{(j^j)})})})}, $$ while $$ k^{khhjj} = ((((k^k)^h)^h)^j)^j, $$ so this probably accounts for the confusion.

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