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Embodiments of the present disclosure generally relate to the recovery and extraction of rare earth elements. More specifically, embodiments of the disclosure relate to methods for separating rare earth elements from coal, coal by-product(s), and/or coal-derived product(s). In an embodiment, a method of removing rare earth elements from a coal-derived product is provided. The method generally includes introducing supercritical CO2 to the coal ash to form a first mixture, introducing a first acid to the first mixture to form a second mixture, and removing a first composition from the second mixture, the first composition comprising the one or more rare earth elements.