We can define 2^{3^{4}} either as (2^{3})^{4} or as
2^{(3^{4})} . Does it make any difference?
Now calculate \left(\sqrt 2^{ \sqrt 2 }\right)^{ \sqrt 2 } and
\sqrt 2 ^{\left(\sqrt 2 ^{ \sqrt 2 }\right)} and answer the
following question for the natural extension of both
definitions.
Which number is the biggest \sqrt 2 ^{\sqrt 2 ^{\sqrt 2 ^{\sqrt
2 ^{.^{.^{.}}}}}}
where the powers of root 2 go on for ever, or \left(\sqrt 2
^{\sqrt 2 }\right)^{\sqrt 2} ?