We can define

2^{3^{4}}

either as

(2^{3})^{4}

or as

2^{(3^{4})}

. Does it make any difference?

Now calculate ${(\sqrt{(} 2)^{\sqrt{(} 2)})}^{\sqrt{(} 2)}$ and $\sqrt{(} 2)^{(\sqrt{(} 2)^{\sqrt{(} 2)})}$ and answer the following question for 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

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