Why 2 ^^ 1 ^^ 2 = 2?

[Ivan Kazmenko]

On Thursday, 26 October 2017 at 10:02:54 UTC, Kagamin wrote:
> On Sunday, 22 October 2017 at 22:28:48 UTC, Ivan Kazmenko wrote:
>> Yeah, and a height-3 tower $a^{b^c}$ (TEX notation)
>
> Is $a^{b^c}$ the same as ${a^b}^c$ ? They are drawn slightly 
> differently, so I suppose it's ambiguous indeed.

Surely not the same.

"3 to the power of (3 to the power of 3)" is "3 to the power of 
27", or 7,625,597,484,987.
"(3 to the power of 3) to the power of 3" is "27 to the power of 
3", or 2187.

For an argument, the TEX command "^" accepts either a single 
character or a bracket-enclosed string of arbitrary length.  So 
$3^3^3$ indeed transforms to ${3^3}^3$, but not for some deeper 
reason this time.

Ivan Kazmenko.