Root Mean Squared Error (RMSE) - significance of square root
It depends on what you are using the RMSE for. If you are merely trying to compare two models/estimators, then there is no significance to the square root. However, if you are trying to plot the error in terms of the same units as you made the measurements/estimates, then you need to take the square root to transform the squared units to the original units (much like variance vs standard deviation)