To find the angular frequencies where the power is half of the maximum power (resonant power), we use the resonant frequency ($\omega_0$) and the half-power bandwidth ($\Delta \omega$).

1. Resonant angular frequency ($\omega_0$): $\omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{5.0 \times 80 \times 10^{-6}}} = \frac{1}{\sqrt{400 \times 10^{-6}}} = \frac{1}{2 \times 10^{-2}} = 50$ rad/s.
2. Half-bandwidth ($\Delta \omega$): The frequencies at which power is halved occur at $\omega = \omega_0 \pm \Delta \omega$, where $\Delta \omega = \frac{R}{2L}$. Substituting the values: $\Delta \omega = \frac{40}{2 \times 5.0} = 4$ rad/s.
3. Calculation: The required angular frequencies are $\omega_1 = 50 - 4 = 46$ rad/s and $\omega_2 = 50 + 4 = 54$ rad/s.