# 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 $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$. This may be able to see through easily, but how can we denest such a complicated one $\sqrt{61-24\sqrt{5}}(=4-3\sqrt{5})$? And Is there any ways to judge if a radical in $\sqrt{a+b\sqrt{c}}$ form can be denested?

Mr. Srinivasa Ramanujan even suggested some CRAZY nested radicals such as:

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.

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 $$subtract out √norm, then divide out √trace\rm \ \ \color{blue}{subtract\ out}\ \sqrt{norm}\:,\ \ then\ \ \color{brown}{divide\ out}\ \sqrt{trace}$$

Recall $$w=a+b√n\rm\: w = a + b\sqrt{n}\:$$ has norm $$=w⋅w′=(a+b√n) ⋅(a−b√n) =a2−nb2\rm =\: w\:\cdot\: w' = (a + b\sqrt{n})\ \cdot\: (a - b\sqrt{n})\ =\: a^2 - n\, b^2$$

and,  furthermore, $$w\rm\ w\:$$ has  trace $$=w+w′=(a+b√n)+(a−b√n)=2a\rm\: =\: w+w' = (a + b\sqrt{n}) + (a - b\sqrt{n})\: =\: 2a$$

Here $$61−24√5\:61-24\sqrt{5}\:$$ has norm $$=292.= 29^2.\:$$ $$subtracting out √norm =29 \rm\, \color{blue}{subtracting\ out}\ \sqrt{norm}\ = 29\$$ yields $$32−24√5\ 32-24\sqrt{5}\:$$

and this has $$√trace=8, thus, dividing it out\rm\ \sqrt{trace}\: =\: 8,\ \ thus,\ \ \ \color{brown}{dividing \ it \ out}\,$$ of this yields the sqrt: $$±(4−3√5).\,\pm( 4\,-\,3\sqrt{5}).$$

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