We present new spatially resolved observations of the dust thermal emission at 7 mm from the Fomalhaut debris disk obtained with the Australia Telescope Compact Array. These observations provide the longest wavelength detection of the Fomalhaut debris disk to date. We combined the new data to literature sub-mm data to investigate the spectral index of the dust thermal emission in the sub-millimeter and constrained the q-slope of the power-law grain size distribution. We derived a value for q = 3.48 ± 0.14 for grains with sizes around 1 mm. This is consistent with the classical prediction for a collisional cascade at the steady-state. The same value cannot be explained by more recent collisional models of planetesimals in which either the velocity distribution of the large bodies or their tensile strength is a strong function of the body size.