The following contradiction has been bothering me for a day or so and I'm sure I'm missing something obvious. You may have seen the infinite power tower problem:
$$ \Large x^{x^{x^{.^{.^.}}}} = 2 $$ $$ \Large x^2 = 2 $$ $$ \Large x = \sqrt{2} $$
Plugging back in the original equation, we have found:
$$ \Large \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{.^{.^.}}}} = 2 $$
However, we also have:
$$ \Large \left( \sqrt{2} ^ {\sqrt{2} }\right)^ \sqrt{2} = \left( \sqrt{2} \right) ^{\sqrt{2} \cdot \sqrt{2}} = \left( \sqrt{2} \right) ^{2} = 2 $$
Therefore
$$ \Large \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{.^{.^.}}}} = \left( \sqrt{2} ^ {\sqrt{2} }\right)^ \sqrt{2} = 2$$
Which seems like a contradiction. Are one or both of my premises false? Thanks.