How to calculate Absolute error under square root sign?
A slightly different approach to @Zhe's answer is to use the 'propagation of errors' formula. Your formula is $y=\sqrt{x}$  with $x=15.13$ and an error $\sigma_x=0.005$ then the formula becomes $\displaystyle \sigma_y=\frac{\mathrm dy}{\mathrm dx}\sigma_x = \frac{\sigma_x}{2\sqrt{x}}= 6.4\cdot 10^{-4}$.

[if you have a formula in more than one variable , say $x$ and $z$ then $y=f(x,z)$ and the equation is more complicated but still quite easy if done in steps and is $\displaystyle \sigma_y^2=\left(\frac{\mathrm dy}{\mathrm dx} \right)^2_z\sigma^2_x+ \left(\frac{\mathrm dy}{\mathrm dz} \right)^2_x\sigma^2_z$. Notice that the square of the final error is calculated in this case.]