The value of \((0.04^{log_{\sqrt{5}}(\frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ...)})\) is

The expression equals $16$. The geometric series sums to $\frac{1/4}{1-1/2}=\frac12$, and $0.04=(\sqrt5)^{-4}$, so the expression is $(\frac12)^{-4}$. Using exponent $-2$ for the base would give $4$.