What is the correct value of 2^5^4^3?

Question

My calculator and google calculated it differently, why? To avoid confusion, where should I put the parenthesis. This can be write in at least 16 ways with parenthesis, but which one gives the correct value?

Answer 1

There is no universally accepted way of placing the parentheses. The expression has no well-defined "correct" value, as opposed to $2-5-4-3 = -10$, say.

Answer 2

There are two interpretations for a^b^c: $a^{(b^c)}$ and $({a^b})^c$. The second one is just $a^{bc}$ and so the usual interpretation is the first one: a^b^c=a^(b^c).

Answer 3

Since $\left(a^b\right)^c = a^{(bc)}$, most mathematicians prefer to use $a^{b^c}$ to represent $a^{\left(b^c\right)}$

so using this you might expect $2^{5^{4^3}}$ to represent $2^{\left(5^{\left(4^3\right)}\right)}$ rather than $\left(\left(2^5\right)^4\right)^3$, giving a result more than $10^{\left(10^{44}\right)}$ rather than a result of $2^{60} \approx 10^{18}$

Different calculators may produce different answers, depending on how they are programmed. Indeed the same calculator in standard mode may say $1+2 \times 3 = 9$ while in scientific mode may say $1+2 \times 3 = 7$