Strategies to denest nested radicals √a+b√c\sqrt{a+b\sqrt{c}}

I have recently read some passage about nested radicals, I’m deeply impressed by them. Simple nested radicals 2+2,322 which the later can be denested into 12. This may be able to see through easily, but how can we denest such a complicated one 61245(=435)? And Is there any ways to judge if a radical in a+bc form can be denested?

Mr. Srinivasa Ramanujan even suggested some CRAZY nested radicals such as:
321,328327,3534,3cos2π7+3cos4π7+3cos8π7,6732019,...
Amazing, these all can be denested. I believe there must be some strategies to denest them, but I don’t know how.

I’m a just a beginner, can anyone give me some ideas? Thank you.

Answer

There do exist general denesting algorithms employing Galois theory, but for the simple case of quadratic algebraic numbers we can employ a simple rule that I discovered as a teenager.


Simple Denesting Rule   subtract out norm,  then  divide out trace

Recall w=a+bn has norm =ww=(a+bn) (abn) =a2nb2

and, furthermore,  w has trace =w+w=(a+bn)+(abn)=2a


Here 61245 has norm =292. subtracting out norm =29  yields  32245

and this has  trace=8,  thus,   dividing it out of this yields the sqrt: ±(435).


See here for a simple proof of the rule, and see here for many examples of its use.

Attribution
Source : Link , Question Author : JSCB , Answer Author : Bill Dubuque

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