Final answer:
Radio waves and gamma-ray emissions provide evidence for quasars being the active nuclei of distant galaxies, confirmed by Hubble Space Telescope observations and high redshifts matching underlying galaxies.
Step-by-step explanation:
The evidence that supports quasars being the nuclei of very distant galaxies predominantly includes radio waves and other forms of electromagnetic radiation. Early quasars were discovered because of their strong radio emissions, and further observations with the Hubble Space Telescope confirmed that quasars are indeed located at the centers of galaxies. Additionally, quasars have been observed to emit gamma-ray radiation from jets shooting out at close to the speed of light. Together with the observable high redshifts in their spectra, which match the redshifts of the underlying galaxies, these forms of radiation provide strong evidence for quasars being extremely luminous and energetic galactic nuclei, often associated with supermassive black holes.