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**rgep** It's quite possible but you have to think a little carefully about what you mean by $\displaystyle 2^{3^4}$, which I'll write on one line as 2^3^4. Do you mean (2^3)^4, ie take 2, raise that to the power 3, then raise the result (8) to the power 4, giving 4096; or do you mean 2^(3^4), ie take the 4th power of 3, then take that power (81) of 2 giving 2417851639229258349412352. These are different in general, and a way of saying that is that the ^ operator is not *associative*. You have actually seen this distinction before: addition and multiplication are associative, that is $\displaystyle a+(b+c) = (a+b)+c$ and $\displaystyle a \times (b \times c) = (a \times b) \times c$, whereas subtraction and division are not.

Anyway it wouldn't really matter which way of inserting the brackets you chose, provided that everyone else understood the same thing by it, were it not for one thing. The first way of reading $\displaystyle a^{b^c}$, as $\displaystyle \left(a^b\right)^c$ has another expression, since it is just $\displaystyle a^{(b\times c)}$. Since we don't really need two ways of writing the same thing, it makes sense for everyone to agree that $\displaystyle a^{b^c}$ should mean $\displaystyle a^{\left(b^c\right)}$.

So the short answer is: yes, $\displaystyle a^{b^c} = a^{\left(b^c\right)}$.