# What is the correct value of 2^5^4^3? [duplicate]

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?

## marked as duplicate by kingW3, Community♦Jun 27 '18 at 11:13

• How the correct value is defined? As you notice, $^$ is not associative. – Jack D'Aurizio Jun 27 '18 at 10:36

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).

• The calculator says 2^5^4^3 = (2^5)^4^3 = (2^5^4)^3 = (2^5^4^3) , and also (2^5)^(4^3) = 2^5^(4^3) , but the others 2^(5^4)^3 and 2^(5^4^3) are different. (Computer calculator tends to use parenthesis at bottom whereas google use them at top generally) – Talha ŞAHİN Jun 27 '18 at 10:46

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 $1+2 \times 3 = 7$

• What about Euler's identity, can it be written as e^i^pi or e^pi^i ? – Talha ŞAHİN Jun 27 '18 at 11:03
• @TalhaŞAHİN Use $e^{i\pi}=-1$ if you want to be understood, though $e^{\pi i}=-1$ would not be wrong. Meanwhile $e^{i^{\pi}}$ or $e^{{\pi}^i}$ would be likely to be interpreted as something else – Henry Jun 27 '18 at 11:22

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.

• I disagree, the common way is to work from right to left. This is because any other order simplifies to multiplication. see en.wikipedia.org/wiki/Order_of_operations#Special_cases – Jens Renders Jun 27 '18 at 10:40
• @JensRenders Literally the second sentence in that link says "However, some computer systems may resolve the ambiguous expression differently." which both says it's ambiguous, and that both standards are in use. – Arthur Jun 27 '18 at 10:43
• Yes and the first sentence is "the usual rule is to work from the top down", and also "because exponentiation is right-associative in mathematics". That is commonly accepted (yes not universally, good edit). Clearly some computer systems implement it differently, hence OP's question. In mathematics though, it is commonly accepted to be right associative for the reason I gave. – Jens Renders Jun 27 '18 at 10:50