I have recently read some passage about nested radicals, I’m deeply impressed by them. Simple nested radicals √2+√2,√3−2√2 which the later can be denested into 1−√2. This may be able to see through easily, but how can we denest such a complicated one √61−24√5(=4−3√5)? And Is there any ways to judge if a radical in √a+b√c form can be denested?
Mr. Srinivasa Ramanujan even suggested some CRAZY nested radicals such as:
3√√2−1,√3√28−3√27,√3√5−3√4,3√cos2π7+3√cos4π7+3√cos8π7,6√73√20−19,...
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+b√n has norm =w⋅w′=(a+b√n) ⋅(a−b√n) =a2−nb2
and, furthermore, w has trace =w+w′=(a+b√n)+(a−b√n)=2a
Here 61−24√5 has norm =292. subtracting out √norm =29 yields 32−24√5
and this has √trace=8, thus, dividing it out of this yields the sqrt: ±(4−3√5).
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
