The frequency of oscillation of a resonant LC circuit is given by the formula $f = \frac{1}{2\pi\sqrt{LC}}$ . Given the initial frequency $f = \frac{1}{2\pi\sqrt{LC}}$. New Inductance $L' = 2L$ and new Capacitance $C' = 4C$. The new frequency $f'$ is given by: $f' = \frac{1}{2\pi\sqrt{L'C'}} = \frac{1}{2\pi\sqrt{(2L)(4C)}} = \frac{1}{2\pi\sqrt{8LC}}$ $f' = \frac{1}{\sqrt{8}} \cdot \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\sqrt{2}} f$.