The mid points of two small magnetic dipoles of length $$d$$ in end-on positions, are separated by a distance $$x$$ ($$x \gg d$$). The magnitude of force between them is proportional to $$x^{-n}$$ where n is:

The magnetic field $$B$$ produced by the first dipole at the center of the second dipole along its axis is $$B = \frac{\mu_0}{4\pi} \frac{2M_1}{x^3}$$

$$U = -\vec{M_2} \cdot \vec{B} = -M_2 B \cos(0^\circ) = -\frac{\mu_0}{4\pi} \frac{2M_1 M_2}{x^3}$$

$$F = -\frac{dU}{dx} = -\frac{d}{dx} \left( -\frac{\mu_0}{4\pi} \frac{2M_1 M_2}{x^3} \right)$$

$$F = \frac{\mu_0}{4\pi} (2M_1 M_2) \frac{d}{dx} (x^{-3})$$

$$F = \frac{\mu_0}{4\pi} (2M_1 M_2) (-3x^{-4})$$

$$|F| \propto x^{-4}$$

$$n = 4$$