The first method selected for extraction of plutonium was a precipitation process called the bismuth phosphate process, developed and tested at the Oak Ridge National Laboratory (ORNL) between 1943 and 1945 to produce quantities of plutonium for evaluation and use in the US weapons programs. The bismuth phosphate process was first operated on a large scale at the Hanford Site, in the later part of 1944. It was successful for plutonium separation in the emergency situation existing then, but it had a significant weakness: the inability to recover uranium. After decladding, the uranium metal was dissolved in nitric acid. The plutonium at this point is in the +4 oxidation state. It was then precipitated out of the solution by the addition of bismuth nitrate and phosphoric acid to form the bismuth phosphate. The plutonium was coprecipitated with this. The bismuth phosphate process is an obsolete process that adds significant unnecessary material to the final radioactive waste.
The first successful solvent extraction process for the recovery of pure uranium and plutonium was developed at ORNL in 1949. The PUREX (Plutonium and Uranium Recovery by Extraction) process is the current method of extraction. The PUREX process is a liquid-liquid extraction method used to reprocess spent nuclear fuel, in order to extract uranium and plutonium, independent of each other, from the fission products. This is the most developed and widely used process in the industry at present. The PUREX process was invented by Herbert H. Anderson and Larned B. Asprey at the Metallurgical Laboratory at the University of Chicago, as part of the Manhattan Project under Glenn T. Seaborg; their patent "Solvent Extraction Process for Plutonium" filed in 1947, mentions tributyl phosphate as the major reactant which accomplishes the bulk of the chemical extraction. Reprocessing (using PUREX or slightly modified form) of civilian fuel has long been employed at the COGEMA La Hague site in France, the Sellafield site in the United Kingdom, the Mayak Chemical Combine in Russia, and at sites such as the Tokai plant in Japan, the Tarapur plant in India, and West Valley Reprocessing Plant, Savannah River Site, Idaho National Laboratory, Oak Ridge National Laboratory, in the United States.
There is no unequivocal answer to this question. The choice of the separation method depends on what element you are going to determine and in what material. Each case requires a specific approach.
It depends, but solvent extraction has many advantages over precipitation, mainly working with liquid phases and so avoiding cumbersome centrifugation, filtration etc. So I would prefer extraction method.