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Does taking the power goes from left to right or from right to left? For example, what is 2^2^3?

According to a textbook I was reading, all operations are executed from left to right, which would make me think that 2^2^3 =$(2^2)^3=4^3$. However, my calculator says it is equal to $2^{(2^3)}=2^8=256$.

Kasper
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  • See https://en.wikipedia.org/wiki/Exponentiation#Identities_and_properties. – lhf Nov 28 '17 at 17:16
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    Without additional context or parentheses to specify, it is common convention to read it from right to left. Your calculator surprisingly got it correct. It is not uncommon for calculators to get this specific question wrong. – JMoravitz Nov 28 '17 at 17:18
  • Since exponentiation is not associative, you could decide to depend on some kind of international agreement to always mean "left to right" or "right to left". Given the manner in which international agreements are rapidly falling apart, I would not advise this. Instead, put the parentheses in to clarify what you mean, or add words to say how you wish to associate. – Lee Mosher Nov 28 '17 at 17:18
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    Exponent towers are always evaluated from right to left, so $a^{b^c}=a^{(b^c)}$. The reason for that definition is that doing it from left to right would be superfluous, because we can always simply rewrite $(a^b)^c=a^{bc}$. – Thorgott Nov 28 '17 at 17:18
  • @LeeMosher it was my understanding that this has been the standard convention for nearly a century. Yes, notations and conventions can and have been known to change over time, but I really doubt that this particular convention will see any change within our lifetimes. – JMoravitz Nov 28 '17 at 17:21
  • "all operations are executed from left to right" You know the joke; never use absolutes. I figure $2^{2^3}$ must mean $2^{(2^3)}$, because if they meant $(2^2)^3$ they'd have just written $2^{2*3}$. Many claim that $2^{(2^3)}$ is the convention. Many (myself included) say it's ambiguous. Nobody claims $(2^2)^3$ is the convention. My advice: i) If you see it, assume $2^{(2^3)}$, ii) never use it. Extra clarity is never bad. (Yup, that's another absolute you should never use.) – fleablood Nov 28 '17 at 17:32
  • PARI/GP is a calculator using the usual way to interprete the power tower. This convention is important for very big numbers. If we have a large tower, the difference of the results is huge, if we calculate once from the left and once from the right. – Peter Nov 28 '17 at 18:06
  • @JMoravitz I agree with all of the arguments on this page, and the linked pages, that the convention should be association from right to left. But the various expressions of confusion amongst some of us on this site, as well as amongst our mechanical devices and the international agreements that are supposed to govern their operations, convince me that this convention is not as standard as one might wish it would be. Hence my advice: be clear about what how one associates when one uses iterated exponentials. – Lee Mosher Nov 28 '17 at 19:16

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Calculators sometimes get confused with operations. For example $6\div 2(2+1)=1$ and $6\div 2(2+1)=9$. Which still brakes the internet to this day... [Different calculators give different results]

Your expression 2^2^3 can be missunderstood. It is not clear what you mean. Going after common convention you "mean" $2^{2^3}=2^{8}=256$

That means ^ notes a "right associative operator". For example "-" notes a "left associative operator", since 1-2-3 is read from left to right, after convention.

Cornman
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