Radical Radicals

Algebra Level 5

Let \(S_n\) be the set of numbers \[\underbrace { \pm \sqrt { 2\pm \sqrt { 2\pm \sqrt { 2\pm \sqrt { 2\cdots \pm \sqrt { 2 } } } } } }_{ n \text{ 2's} }. \] For instance, \[S_2=\left\{ \sqrt { 2+\sqrt { 2 } },\ \sqrt { 2-\sqrt { 2 } },\ -\sqrt { 2+\sqrt { 2 } },\ -\sqrt { 2-\sqrt { 2 } } \right\}. \]

What is the value of

\[ \large \prod_{a \in S_{2017} } ( 1 - a ) ? \]


Inspiration.

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