To see this, let us choose the periods of two definite planets - say, Earth and Jupiter. For these the equation assumes the following form ('J' and 'E' indicating 'Jupiter' and 'Earth' respectively):
r_J³ / r_E³ = (t_J / t_E) / (t_E / t_J)
Let us now see what meaning we can attach to the two expressions
t_J / t_E and t_E / t_J.