TITLE: Incorrect notation in math?
QUESTION [7 upvotes]: 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? $$\sqrt[\sqrt{2^3}]{2}$$

REPLY [3 votes]: $a^{\large b^{\Large c}}$ 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.