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Squared means to times something by itself. Every mathematician's fave trick for taking things that can be negative and turning positive.
Mean is the arithmetic average. Integral of x f(x) dx
Root is the square root, takes the awkward squared values back to normal.
Sound and its structure: it's made of waves. So an amp or speaker deals with periodic signal. RMS is a way of quantifying power averaged over the waves.
Kind of like putting Peak power, or current draw on the back of an amp... and not saying it is current draw/whatever. So then, when joe or stefan or mark or who ever go to buy an amp, they only see some bogus four digit number, and assume that is the output power.