Is 1^2^3 = $1^{2^3}$ or $(1^2)^3$ 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 $1^3$?
 A: 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.
A: 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
A: 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.
A: 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.
A: 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.
