  The 'spectral age problem' is our systematic inability to reconcile the
maximum cooling time of radiating electrons in the lobes of a radio galaxy with
its age as modelled by the dynamical evolution of the lobes. While there are
known uncertainties in the models that produce both age estimates, `spectral'
ages are commonly underestimated relative to dynamical ages, consequently
leading to unreliable estimates of the time-averaged kinetic feedback of a
powerful radio galaxy. In this work we attempt to solve the spectral age
problem by observing two cluster-centre powerful radio galaxies; 3C320 and
3C444. With high-resolution broad-band Karl G. Jansky Very Large Array
observations of the radio sources and deep XMM-Newton and Chandra observations
of their hot intra-cluster media, coupled with the use of an analytic model, we
robustly determine their spectral and dynamical ages. After finding
self-consistent dynamical models that agree with our observational constraints,
and accounting for sub-equipartition magnetic fields, we find that our spectral
ages are still underestimated by a factor of two at least. Equipartition
magnetic fields will underestimate the spectral age by factors of up to ~20.
The turbulent mixing of electron populations in the radio lobes is likely to be
the main remaining factor in the spectral age/dynamical age discrepancy, and
must be accounted for in the study of large samples of powerful radio galaxies.
