homework and exercises - Deriving the speed of the propagation of a change in the Electromagnetic Field from Maxwell's Equations

I've been told that, from Maxwell's equations, one can find that the propagation of change in the Electromagnetic Field travels at a speed $\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ (the values of which can be empirically found, and, when plugged into the expression, yield the empirically found speed of light)

I'm really not sure how I would go about finding $v = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$ simply from Maxwell's equations in the following form, in SI units --

$$\nabla \cdot \mathbf{E} = \frac {\rho} {\epsilon_0}$$

$$\nabla \cdot \mathbf{B} = 0$$

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$

$$\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right)$$

Is what I believe true? (that the speed of propagation is derivable from Maxwell's Equations)

If not, what else is needed?

If so, can you provide a clear and cogent derivation?