When we raise a signal to a power n, does the bandwidth always gets multiplied by n?
If you raise your time-domain signal to the power n, then your bandwidth does increase, but not necessarily by a factor of n. It will depend upon how you want to define bandwidth.

This is because multiplication in the time domain amounts to convolution in the frequency domain. Let's take the example of n=2. If we denote the Fourier transform operator to be \$F\$, then

$$
F[y^2(t)]= \int Y(\omega')Y(\omega-\omega' )d\omega'
$$

You will have non-zero spectral content whenever \$Y\$ overlaps with a shifted version of itself. Therefore your signal has spectral content covering a region twice as wide as that of the original signal, but the 3 dB point may not necessarily double. So it depends a bit on how you want to define bandwidth. For a Gaussian time domain pulse, I believe the 3-dB bandwidth increase will go like \$n^{1/2}\$. The graphic below might help illustrate the point:



(picture borrowed for illustrative purposes from here)

Each time you multiply again by \$y(t)\$ in the time domain, you convolve again with \$Y(\omega)\$, increasing the bandwidth yet again. 

The other effect of raising the time-domain signal to a power can be described as generating \$\textit{harmonics}\$ of the original signal. This is best visualized with a signal that has somewhat local support near its center frequency rather than extending from DC on upward (i.e. a 'narrow band' signal). In that case, the spectral content at negative frequency and the spectral content at positive frequency produce, in the convolution, content centered at the sum and difference frequencies (of the center frequency). Each of those 'harmonics' will have a bandwidth larger than the original bandwidth due to the spreading introduced by the convolution. 

You could legitimately view these harmonics as an increase in the bandwidth well beyond the the effect described so far because you have spectral content all the way out to the harmonics. So I suppose it depends a bit on what is important to you, and what you will do with the signal after raising it to the nth power. If you intend to filter out harmonics, then you should look at raising to the nth power as increasing the bandwidth by some factor like \$n^{1/2}\$, depending on how you define bandwidth. If you aren't going to filter, then you should probably consider the bandwidth to be approximately n times the center frequency that your original signal contained for a 'narrow band' signal due to the generation of harmonics.