Not sure I follow your question, but Smart$ points out an imprecision in my earlier comments. So let me expand on my earlier comments:
Say you have a function of time over some interval [f(t) defined on the interval (0,T)], such as one cycle of a periodic function, and you'd like to know the average value of the function. One obvious way to do this is to take the actual average, i.e. pick some (equally) distributed points in time, find the function values at those points in time, and average the function values [more precisely, average f = 1/T * integral(f(t)dt)].
So you can do this for a current waveform I, and for a voltage waveform V, and for the power waveform P = I * V. In DC, where I and V are contstant functions of time, then you get (average P) = (average I) * (average V). But in AC, that won't work, as average I = average V = 0 (assuming no DC component). In general, the average of the product is not the product of the averages.
In the case of the power waveform, the average we want really is the usual average described above. We just need to a new notion of average for current and voltage so that (average P) = (new_average I) * (new_average V). When the I and V waveforms have the same shape and phase (i.e. V = alpha * I for some constant alpha), then the notion of average for I and V that we need is root mean square, RMS. So we have (average P) = (RMS I) * (RMS V).
Cheers, Wayne
