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Caret ^ signs can be used to describe the power of numbers.

Is $1$^$2$^$3 = 1^{(2^3)}$ or $(1^2)^3$

How do you calculate it? Do you start with $2^3$ and then do $1^8$ or do you start with $1^2$ and then do $1^3$?

TonyK
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    This is just bad notation. – Ali Caglayan Oct 24 '14 at 15:28
  • @Alizter What do you mean? –  Oct 24 '14 at 15:30
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    a^b^c is generally interpreted as $a^{(b^c)}$ since $(a^b)^c$ is equal to $a^{bc}$. – Thomas Andrews Oct 24 '14 at 15:34
  • @Murplyx While what Thomas says is true, it is just easier to read a^(b^c) than to read a^b^c and remember that (a^b)^c would be more compactly written a^bc (and hope that whoever wrote a^b^c also realised this!). This is probably what Alizter was getting at - while there might be a sensible convention for interpreting a^b^c, it doesn't come up so often, so it's nice to help people out with some brackets. – mdp Oct 24 '14 at 15:46
  • @graydad, your edit rather missed the point of the question! – TonyK Oct 24 '14 at 16:42
  • @Murplyx Answer this and I'll answer your question –  Oct 24 '14 at 16:53

5 Answers5

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Parenthesis are essential in this situation. Without them, we can't make sense of the expression. Not everyone will understand the same thing of it, a priori. We have: $$1^{\left(2^3\right)} = 1^8, \quad \left(1^2\right)^3 = 1^3.$$ This can lead to serious problems, such as a beginner in calculus making a confusion between $e^{-t^2}$ and $(e^{-t})^2$, etc.

Ivo Terek
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  • So you mean that it can be recognized as both if no parentheses are specified? –  Oct 24 '14 at 16:05
  • The expression alone just doesn't make sense. As Alitzer said in the comments, this is just a bad notation. – Ivo Terek Oct 24 '14 at 16:07
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When you work with nested exponents like this the order of evaluation starts at the top and moves down. In the context of your example, that means you first evaluate $2^3=8$ and second evaluate $1^8 = 1$. For further reading I suggest perusing a page from the source of all knowledge, http://en.wikipedia.org/wiki/Tetration

graydad
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    If no parenthesis are written as graydad says you compute top$\rightarrow$down. If at any point you intend the order to be done differently, you may use parenthesis. Of course, using extra parenthesis never hurt anyone, so I usually write $3^{3^3}$ as $3^{(3^3)}$ anyways – JMoravitz Oct 24 '14 at 15:34
  • @JMoravitz good point; thank you for pointing that out! – graydad Oct 24 '14 at 15:37
  • Tetration is a terrible reference since it only refers to cases where all the terms are the same. – Thomas Andrews Oct 24 '14 at 15:39
  • @ThomasAndrews You are right; this isn't tetration, but I think "terrible" is a bit harsh to say. It was the only wikipedia page I could find that had a portion discussing nested exponentials. Learning how tetration works should at least give OP an idea of how to evaluate nested exponentials. – graydad Oct 24 '14 at 15:42
  • @JMoravitz Extra parentheses has hurt people all the time. $abcde$ is much easier to read than $((ab)c)(de)$, and being hard to read is harm. – Thomas Andrews Oct 24 '14 at 15:42
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This is a matter of convention, just as it is not a priori given that $a+b\cdot c$ stands for $a+(b\cdot c)$ instead of $(a+b)\cdot c$ or that $a-b-c$ stands for $(a-b)-c$ and not for $a-(b-c)$. Since the rules of exponentiation show us that $(a^b)^c=a^{bc}$ it would be a waste of notation to let ${a^b}^c$ stand for $(a^b)^c$. Hence the convention is that $a^{b^c}$ stands for $a^{(b^c)}$.

On the other hand, in situations where you are forced to use the caret sign for exponentiation, I would always make the meaning clear with parentheses, namely because the convention is that noncommutative equal operators are evaluated left to right (such as $a-b-c=(a-b)-c$ and $a/b/c=(a/b)/c$) so that "downgrading" exponentiation to such a linear notation bears the risk of people applying that convention to exponentiation as well.

Hagen von Eitzen
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It is standard for exponentiation to be right-associative: that given a chain of repeated exponentiations, you compute them from right to left. i.e.

$$ a^{b^c} = a^{\left(b^c \right)} $$

and it is standard for carat or arrow notation for exponentiation to be read the same way:

$$ a \hat{} b \hat{} c = a \hat{}(b \hat{} c) $$ $$ a \uparrow b \uparrow c = a \uparrow (b \uparrow c) $$

Due to the low frequency to which this comes up in handwritten mathematics, people don't learn to instinctively parse it correctly, and thus it is recommended that you avoid writing chained carats or chained uparrows, and instead use parentheses to make it impossible to misread the intent.

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I'm a little bit confused as to why some would have difficulty with the notation $a^{b^c}$, while at the same time consider $a^{bc}$ to be completely unambiguous. In both cases, prior to evaluating the exponential, the power needs to be evaluated. The fact that the power itself is another exponential expression is irrelevant.

Having said that, where carets are involved it's probably best to insert parentheses.

John Joy
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