Many Square Roots
Source:
April 6, 2008
Problem Statement
Solve the equation
\sqrt {x \plus{} \sqrt {4x \plus{} \sqrt {16x \plus{} \sqrt {\dotsc \plus{} \sqrt {4^{2008}x \plus{} 3}}}}} \minus{} \sqrt {x} \equal{} 1.
Express your answer as a reduced fraction with the numerator and denominator written in their prime factorization.