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## 'Climbing Powers' printed from http://nrich.maths.org/

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 \[ \left(\sqrt 2
^{\sqrt 2 }\right)^{\sqrt 2} ? \]