Radical Radicals

Algebra Level 3

How many positive \(x\) satisfy

\[\Large \sqrt{x} + \sqrt{\sqrt{x}} + \sqrt{\sqrt{\sqrt{x}}} + \ldots + \underbrace{\sqrt{\sqrt{\sqrt{\cdots\sqrt{x}}}}}_{n \text{ square roots}}=1\]

for a given positive integer \(n?\)

Bonus: What sorts of bounds can you put on these solution(s) in terms of \(n?\)

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