Evaluating nested radicals

Calculus Level 1

2220+2221+2222+2223+ \sqrt{ \frac{2}{2^{2^0}} + \sqrt{ \frac{2}{2^{2^1}} + \sqrt{ \frac{2}{2^{2^2}} + \sqrt{ \frac{2}{2^{2^3}} +\cdots }}}}

If the value of above expression is in the form A\sqrt{A}, find the value of AA.

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