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Does $x^{y^z}$ equal $x^{(y^z)}$? If so, why?

Why not simply apply the order of the operation from left to right? Meaning $x^{y^z}$ equals $(x^y)^z$?

I always get confused with this and I don't understand the underlying rule. Any help would be appreciated!

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The most important thing to realize is $(x^y)^z \ne x^{(y^z)}$ (just try some values. I actually didn't realize there was a convention that $x^{y^z}$ was defined to be $x^{(y^z)}$ but it makes sense. $(x^y)^z = x^{yz}$ so we don't need another way to express that but $x^{(y^z)}$ doesn't equal anything simpler. It's a little like "why" $a*b + c = (a*b) + c$ and not $a*(b + c)$. It could have equalled $a*(b + c)$ because we have a distributive law $a*(b + c) = (a*b) + (a*c)$ so the need to express $(a*b) + (a*c)$ simply as $ab + ac$ seemed more ...to be continued ... – fleablood Jan 30 at 23:39
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... cont. ... It could have equalled $a*(b + c)$ because we have a distributive law $a*(b + c) = (a*b) + (a*c)$ so the need to express $(a*b) + (a*c)$ simply as $ab + ac$ seemed more necessary then the need to express $a*(b + c)$ as $a*b + c$. In any event, if clarity matters just put in the parenthesis. – fleablood Jan 30 at 23:41
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It's just one of those conventions that people have adopted. This one saves on brackets as we can write $a^{b c}$ for $(a^b)^c$, and $a^{b^c}$ for $ a^{(b^c)}$ – user254665 Jan 31 at 1:48
    
One possible motivation I think for the convention is that $\exp \exp x$ can only reasonably be interpreted as $\exp ( \exp x)$ (where $\exp x$ is a common notation for the ubiquitous $e^x$). Choosing $x^{(y^z)}$ over $(x^y)^z$ would keep to that. – Vandermonde Feb 1 at 0:43
    
@fleablood: (1) I see no indication that Imre believes that ${(x^y)}^z = x^{(y^z)}$; the question is about interpreting ${x^y}^z$. (2) Your point about ${(x^y)}^z$ equaling $x^{yz}$ is good, but Rob Arthan made it a few minutes before you did. (3) ISTM that your $a*b+c$ analogy isn't particularly relevant: $*$ and $+$ are distinct operators; $*$ has a higher precedence than $+$; case closed. A more interesting analogy would be $a \div b \div c$, which is generally interpreted as $(a \div b) \div c$, because division (and subtraction) are left-associative, in contrast to exponentiation. – Scott Feb 2 at 0:21
up vote 57 down vote accepted

In the usual computer science jargon, exponentiation is right-associative, which means that $x^{y^z}$ should be read as $x^{(y^z)}$, not $(x^y)^z$. One way to remember this is that $(x^y)^z = x^{yz}$, so it would be silly if out of the two possibilities, $x^{y^z}$ meant the one that can be expressed without using two tiers of superscripts.

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Nice KIS(S) mnemotecnic! – JnxF Jan 30 at 23:24
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Though, $x^{y^z}$ isn't ambiguous at all... it's just the version without formatting ($x^y^z$) that needs a rule about which operation to apply first. (I notice that MathJax actually treats that construction as ambiguous.) – Brilliand Jan 31 at 4:05
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MathJax is presumably imitating $\TeX$'s double subscript error. $x^{{\mbox{$y$}}^{\mbox{$z$}}}$ is ambiguous if you and your typesetter haven't conspired to make the superscripts shrink. – Rob Arthan Jan 31 at 11:09

Usually, a^b^c is taken to mean a^(b^c). This is purely an issue of the definition of notation so deep "why" answers aren't super likely. The main thing is that we have the identity (for positive $a$): $$(a^b)^c=a^{bc}$$ so it would make little sense to make that the default order, given that it reduces to a simpler form, whereas $a^{(b^c)}$ doesn't reduce. Moreover, generally exponentiation is written as $a^{b^c}$ rather than a^b^c, and the former notation more clearly shows that all of $b^c$ is in the exponent.

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Not usually. Always. – EJP Jan 31 at 23:46
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@EJP Not in situations where you're limited to ASCII and have little to no typesetting capabilities (or it is a great deal of trouble to get around these limitations), a situation that you might find surprisingly common outside of tools specifically designed for math. – jpmc26 Feb 1 at 8:24

The notation helps here; the exponent (which is the part that's raised) always acts like it has parentheses around it. So $x^{y^z}$ means $x^{(y^z)}$. Similarly, $x^{y+z}$ means $x^{(y+z)}$ and $x^{yz}$ means $x^{(yz)}$, even though exponentiation has higher precedence than addition or multiplication (so $x+y^z$ means $x+(y^z)$ and $xy^z$ means $x(y^z)$).

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Without a convention, $x^{y^z}$ might be interpreted as either $(x^y)^z$ or $x^{(y^z)}$; so a convention is useful. If the convention meant the first, then we would be obliged to use parentheses whenever we intend the second. On the other hand, if the convention means the second (which it does), then there is no need to write parentheses for the first, because it can anyway be written more simply as $x^{yz}$. The convention predates computer code, and was adopted to save writing lots of parentheses.

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I would just like to point out that many calculators share the OP's confusion, even calculators from the same manufacturer. Taking a quick sample from the lost-and-found box in my office, I found that 2^3^4 turned out to be:

  • 4096 on Texas Instruments BA II Plus, TI-30XA, TI-30X II s, TI-36X solar, Windows calculator
  • 2.4178...*10^24 on Texas Instruments TI-30XS MultiView, Casio fx-115ES Plus, Google search bar.
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No BS here, excellent use of time! +1 – Bacon Feb 3 at 14:36

1) Compute $y^z$ which will be some number $a$

2) Do $x^a$

Since generally $y^z \neq x^y$ you can easily understand why it's not the same.

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