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 $\left(\sqrt 2 ^{\sqrt 2 }\right)^{\sqrt 2} ?$