Ramanujan was a master at manipulating radicals. Some of his unusual ones are
3√5√325−5√275=5√125+5√325+5√−925 √5√15+5√45=−5√1125+5√2125+5√8125+5√16125 4√3+24√53−24√5=4√5+14√5−1 8√1+√1−(−1+√52)24=−1+√521+4√5√2 The pattern of (2) is certainly suggestive, though I haven't seen any generalization of this. And (4) is the 8th root of an expression involving a 24th power! Also, note that he only uses ratios of √pk with the small primes p=2,3,5. Either such simple relations are possible only for these, or Ramanujan found the low-hanging fruits and there are similar ones with prime p>5.
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