Iterated Roots with Powers

Algebra Level 3

\[\sqrt{ 2^1+ \sqrt{ 2^2+ \sqrt{ 2^4 + \sqrt{2^8 + \cdots } } } } \]

The value of above expression is in the form \(\phi \sqrt{A} \), where \(A\) is an integer. Find \(A\).

Notation: The symbol \(\phi\) denotes the golden ratio, \(\phi = \dfrac{1+\sqrt5}2\).

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