TITLE: Why can't we remove the sqrt from rms?
QUESTION [17 upvotes]: In chemistry, we define the root-mean-square speed as
$\sqrt{\bar{u^2}}$ = $\sqrt{\frac{3\text{RT}}{\text{M}}}$
A student asked me why we can't just remove the square root symbol. And aside from "because this is how we define it", I didn't actually have a reason.
So, I'm hoping someone can shed some light on why the above equation is used and not:
$\bar{u^2} = \frac{3\text{RT}}{\text{M}}$
In case it is important, we use this equation to determine the rms speed of a gas. It depends on the temperature (T) and the molecular mass of the gas (M). R is a constant value. I understand we don't just use the average because in a set of gases, they move in a random direction so the average is 0. But, by squaring isn't that issue resolved, without the square root?

REPLY [2 votes]: The RMS of $u$ is defined by the left side of the equation, for whatever physical quantity $u$ you're considering.  It's just coincidence in this case that the right hand side of the equation, which expresses this in terms of physical parameters of the system, also has a square root.  You could find many examples where that's not the case and this particular "simplification" wouldn't make sense.  As other have noted, the "root" and the "square" part of the definition are chosen to ensure that you get an answer in the same units as $u$ itself.