Incorrect notation in math? Does math have an incorrect notation / syntax? I don't mean writing misaligned notation (google), but when you take something like a number to powers to powers to powers, $${{2^2}^2}^3$$ (I was told this is incorrect notation by a teacher). Is it really incorrect, or does it just need to be simplified with parentheses? Do people write maths like this?
a radical expression with the root being a radical expression? $$\sqrt[\sqrt{2^3}]{2}$$
 A: $a^{\large b^{\Large c}}$ means $a^{\large(b^{\Large c}\large)}$. This is not in dispute.
Note that $\left(a^{\large b}\right)^{\large c}=a^{\large bc}$, so there would be no reason for it to mean this.
A: Your teacher is mistaken.  There is a well-established and universal convention about the meaning of an expression like $$2^{2^{2^3}}$$it is always understood to mean $$2^{\left(2^\left(2^3\right)\right)} =2^{2^8} = 2^{256}$$  People can and do write expressions like these.  For example this paper, "Analog of the Skewes Number for Twin Primes",  by Marek Wolf, contains the expressions $$10^{10^{10^{10^3}}}\qquad\text{and}\qquad 10^{10^{529.7}}$$on the first page, with no further explanation.  Similarly "Some Rapidly Growing Functions" by Craig Smoryński  has $$10^{10^{10^{34}}} < e^{e^{e^{e^{4.369}}}}$$ and similar expressions. (I picked these two papers arbitrarily; they were the first two hits in Google Scholar for "Skewes' Number".)
There is a good reason for the convention about what $a^{b^c}$ means:  $a^{b^c}$ could be understood as either $a^\left({b^c}\right)$ or as $\left(a^b\right)^c$.  But 
if it were understood as $\left(a^b\right)^c$, one would never need to write $a^{b^c}$, since it would be equal to $a^{bc}$.  So it is always understood as 
$a^\left({b^c}\right)$.
Nobody ever writes $$\sqrt[\sqrt{2^3}]2$$ even though its meaning is clear. Partly this is because it would have been difficult to typeset with old-fashioned metal type, so there is a tradition of expressing this differently.  And partly it is because it looks bad.
Since by definition, $$\sqrt[a]b = b^{1/a},$$ one would almost always write something like $$(2^{1/2})^{1/2^{3/2}}$$ instead, at which point it would become clear that the expression could be simplified to $$2^{(1/2)(1/2^{3/2})} = 2^{1/2^{5/2}} = 2^{2^{-5/2}}.$$  Good notation enables and encourages this sort of simplification; bad notation obscures and impedes it.
A: Towers of exponents have been standard for ages. Cajori, in his book History of mathematical notations, §313, tells the story. He reproduces an image from a book by Waring published in 1785:

A: I think I'm starting to see the problem; exponentiation is right associative. Perhaps a more sensible notation (for $a^{b^c}$) would be
$${}^{{}^c} {}^b a$$
