Absorption of waves in a plasma Suppose a plasma has characteristic frequency $\omega_p$. Since 
$$n = \sqrt{\left(1-\frac{\omega_p^2}{\omega^2}\right)} $$
For $\omega<\omega_p$, the refractive index will be imaginary - which corresponds to absorption of light.
For $\omega > \omega_p$, the refractive index will be real.
Suppose plasma frequency is $9\: \mathrm{MHz}$. Why is it that AM radio waves ($\sim 1\: \mathrm{MHz}$) are able to be transmitted further than FM radio waves ($\sim 100\: \mathrm{MHz}$)? Isn't that counter-intuitive?
 A: In your question you say that an imaginary refractive index corresponds to absorption of light - that is not true: it just means that the wave electric field decays exponentially. It can be due to absorption but also due to reflection of the wave.
In your example, the plasma frequency is $9\,\mathrm{MHz}$. Any wave injected upon the plasma with a frequency below that value cannot penetrate the plasma. For the example of a radiowave emitted into the direction of the ionosphere (which is a plasma layer), it will be reflected at the ionosphere if the wave frequency is below the plasma frequency of the ionosphere. If the wave frequency exceeds the plasma frequency of the ionosphere (microwaves in the $\mathrm{GHz}$ range, for example, used for communication with satellites), it can penetrate the ionosphere and reach the satellite.
A simple picture to understand this is the following: the plasma frequency is the fastest answer the plasma can give to an external perturbation. When you apply an external electric field, the electrons of the plasma will react to it and screen it. The shortest time scale, the fastest they can react, is the inverse of the electron plasma frequency. An electromagnetic wave with an electric field oscillating slower than the electron plasma frequency is reflected by the plasma, since the electrons cancel the wave electric field. If the wave frequency is larger than the electron plasma frequency, the electron response by the plasma is too slow, the wave can penetrate into the plasma.
I am not sure why you differentiate between AM and FM, I guess that is a typo (since they have both different frequencies). Note that AM stands for Amplitude Modulation and FM for Frequency Modulation. 
A: Your formulas and reasoning do not take into account collisions in plasma or reflections from the ionosphere.
And, by the way, imaginary refraction index does not necessarily corresponds to absorption.
