Iterated Roots with Powers

Algebra Level 3

21+22+24+28+\sqrt{ 2^1+ \sqrt{ 2^2+ \sqrt{ 2^4 + \sqrt{2^8 + \cdots } } } }

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

Notation: The symbol ϕ\phi denotes the golden ratio, ϕ=1+52\phi = \dfrac{1+\sqrt5}2.

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