The dimension of $ \sqrt{\frac{\mu_0}{\epsilon_0}} $ can be understood as follows:

Vacuum permeability ($ \mu_0 $) and vacuum permittivity ($ \epsilon_0 $) relate to the properties of inductance and capacitance, respectively, in a vacuum. We start by considering the formulas for inductance ($ L $) and capacitance ($ C $):

Inductance ($ L $) is given by:

$ L = \frac{\mu_0 \cdot \text{Number of turns} \cdot \text{Area}}{\text{Length}} $

Capacitance ($ C $) is given by:

$ C = \frac{\text{Area} \cdot \epsilon_0}{\text{Distance}} $

From these, the ratio $\frac{L}{C}$ can be expressed as $\frac{\mu_0}{\epsilon_0}$:

$ \frac{L}{C} \propto \frac{\mu_0}{\epsilon_0} $

Taking the square root of this ratio, we have:

$ \sqrt{\frac{\mu_0}{\epsilon_0}} \propto \sqrt{\frac{L}{C}} $

This simplifies further to considering the relationship between time constant ($\tau$) and resistance ($R$):

$ \frac{L}{C} = \frac{\tau R}{(\tau / R)} = R^2 $

Thus, by taking the square root:

$ \sqrt{\frac{\mu_0}{\epsilon_0}} = R $

In conclusion, the dimension of $ \sqrt{\frac{\mu_0}{\epsilon_0}} $ is equivalent to the dimension of resistance, $ R $.