Decibels are decibels. There is no such thing as "root-power decibels".

By definition a decibel (dB) is a ratio of two powers. The formula is 10 * log10(P1/P2) where P1 and P2 are the power of two signals, respectively.

In electronics, however, we usually manipulate and measure voltage levels. It's convenient to represent the ratio of two voltage levels in dB. To do this you would need to square the voltage to get the power (since P = V^2 / R). We also assume R = 1 for convenience. With a little math you get dB = 20 * log10(V1/V2).

Therefore if we reduce the voltage level of a signal by a factor of 0.1 then the signal is now -20 dB relative to before.

dB is simply an easy-to-read logarithmic-to-linear mapping. Music, human perception, and many other things in nature typically have a logarithmic response. The decay of, for example, a cymbal is logarithmic. If you plot this on a linear axis it's hard to display because of the dynamic range. But if you use a logarithmic axis you "compress" the data into something that's easier to view. Decibels are just a widely accepted mapping. You could use any base for the log; log2, ln, etc but since we have 10 fingers log10 is nice.

The point is that X dB is X dB. If you reduce a signal by 20 dB you've reduced it's voltage to 10% of what it was previously. You also reduced it's power to 1% of what it was previously. These are the same things: 20 * log10(0.1) = 10 * log10(0.01).