Why can't we remove the sqrt from rms? 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?

$T$ has units of Kelvin (K). The gas constant $R$ has units of Joule/mole/K. The molecular mass $M$ has units of kg/mole. Also remember that a Joule is $\mathrm{kg.m^2/s^2}$. So the units of $3RT/M$ are $$\mathrm{\frac{kg\,m^2\,K}{s^2\,mole\,K\,kg\,mole^{-1}}=\frac{m^2}{s^2}}$$ which is a velocity squared. So taking the square root gives the correct units for a velocity.