17. $[\mathbf{9}]$ Solve the equation
$$
\sqrt{x+\sqrt{4 x+\sqrt{16 x+\sqrt{\ldots+\sqrt{4^{2008} x+3}}}}} \sqrt{x}=1 .
$$

Express your answer as a reduced fraction with the numerator and denominator written in their prime factorization.
Answer: $\frac{1}{2^{4016}}$ Rewrite the equation to get
$$
\sqrt{x+\sqrt{4 x+\sqrt{16 x+\sqrt{\ldots+\sqrt{4^{2008 x+3}}}}}}=\sqrt{x}+1 .
$$

Squaring both sides yields
$$
\sqrt{4 x+\sqrt{\ldots+\sqrt{4^{2008} x+3}}}=2 \sqrt{x}+1 \text {. }
$$

Squaring again yields
$$
\sqrt{16 x+\sqrt{\ldots+\sqrt{4^{2008} x+3}}}=4 \sqrt{x}+1 .
$$

One can see that by continuing this process one gets
$$
\sqrt{4^{2008} x+3}=2^{2008} \sqrt{x}+1,
$$
so that $2 \cdot 2^{2008} \sqrt{x}=2$. Hence $x=4^{-2008}$. It is also easy to check that this is indeed a solution to the original equation.