Large scale structure will be in the next decade one of the main probes to study cosmology, due to its high information content over a large range of scales and redshifts. Traditional galaxy surveys in the optical wavelengths, like Euclid, Vera Rubin and DESI, are classified into photometric and spectroscropic probes, with the former giving larger statistical power and a good angular resolution and the latter providing a better resolution along the line-of-sight and 3D information. With the advent of SKAO and its revolutionary improvements in radio astronomy, we will be able to measure not only galaxy positions, redshifts and the cosmic shear field, but also the 21cm intensity map (IM) of the brightness temperature of the HI line of neutral Hydrogen, which will provide an independent set of observations on the large scale structures of the Universe with a very precise redshift estimation. Given the synergies between optical and radio observations, we want to estimate in this work how well future experiments will be able to measure deviations beyond our standard cosmological model LCDM, using a parametrized modification of the Poisson equations of GR with two functions \mu and \Sigma that return to unity in the GR-limit. Using Fisher matrix forecasts for spectroscopic galaxy clustering in redshift space, photometric galaxy clustering, weak lensing (including cosmic shear and intrinsic alignments) and IM, including all their non-negligible cross-correlations for optical and radio surveys, we can estimate the accuracy at which we will be able to determine these parametrized deviations from Einstein's gravity. We find that the cross-correlation of IM with DESI, combined with the continuum surveys from SKAO will yield the best constraining power, due to the large range of redshifts probed at different resolutions. Under these settings, the functions \mu and \Sigma measured at z=0, will be determined with 0.7-0.9% accuracy, being competitive with a combination of only stage-IV photometric and spectroscopic surveys,such as DESI+Vera Rubin, but offering independent observations and new degeneracy directions that will be especially important to break the degeneracy between systematic and statistical errors. Despite the still unresolved challenges in the modelling of the nonlinear regime, these forecasts show that the optical-radio synergies will be a very important tool in the next decade to find out what lies beyond our standard cosmological model.
