Does the unit of a quantity change if you take square root of it? For example, I have a mass, m = 0.1kg and I square root it, giving me m = 0.316 (3s.f.) does the unit still stay as kg, or does it change?
 A: Yes, the dimension of a quantity changes if it is square-rooted. If $m$ is a mass with dimension $[m]=\textrm{kg}$, $\sqrt{m}$ is not a mass, but another quantity with dimension $[\sqrt{m}] = \textrm{kg}^{1/2}$.
More generally, if $[a] = A$ and if $[b]=B$, then $[a^n b^m] = A^nB^m$ etc.
A: It becomes the square root of the unit.  Think of energy:
$$E = \frac{1}{2}mv^{2}$$
If I solve for $v$, I have $v = \sqrt{\frac{2E}{m}}$.  Since $\rm 1 J = 1 kg \cdot m^{2}/s^{2}$, we see that the units have to obey the square root, or we will end up with our velocity equalling something other than m/s.
A: Take the root of the unit of area (Eg: 4 m$^2$ )
We get the unit of length (Eg: 2 m)  which is the unit for different physical quantity
So it definitely changes
A: Let's square root 0.1kg:


*

*expressed in kg, we get $\sqrt{0.1}\approx 0.316$.

*expressed in g, we get $\sqrt{100}=10$.


So obviously the unit changes. If it stayed the same, we'd have $0.316\mbox{kg} = 10\mbox{g}$ which is clearly false.
A: As the other answers (and dmckee's comments) note, yes, if you take the square root of a dimensional quantity then you need to take the square root of the units too:
$$ \sqrt{4\;{\rm kg}} = 2\;{\rm kg}^{\frac12} $$
And no, I can't think of any meaningful physical interpretation for the unit ${\rm kg}^{\frac12}$ either.
However, in the comments you say that you were "told to plot a graph of distance against square root of mass."  What that means is simply that you should scale the mass axis non-linearly, presumably in order to more clearly show the relationship between the two quantities.  For labeling the mass axis, you basically have two choices:


*

*label the axis $\sqrt m$, with equally spaced ticks at, say, $1\;{\rm kg}^{\frac12}, 2\;{\rm kg}^{\frac12}, 3\;{\rm kg}^{\frac12}, 4\;{\rm kg}^{\frac12}, \dotsc$, or

*label the axis $m$, with equally spaced ticks at $1\;{\rm kg}, 4\;{\rm kg}, 9\;{\rm kg}, 16\;{\rm kg}, \dotsc$.
While, technically, both of these are valid, I would strongly recommend the latter option.  Just compare these two plots and see which one you find easier to read:

$\hspace{60px}$

Alas, not all plotting software necessarily supports such axis labeling, or at least doesn't make it easy, which is why you sometimes see plots with funny units like ${\rm kg}^{\frac12}$.
