Nested radicals

Nested radicals in Algebra are radical expressions that have another radical expression nested inside a radical. Examples include

:sqrt{5-2sqrt{5}}

which arises in discussing the Pentagon,

:sqrt{5+2sqrt{6}},

or more complex ones such as

:sqrt[3]{2+sqrt{3}+sqrt[3]{4}}.

Denesting these radicals is generally considered a difficult problem. A special class of nested radical can be denested by assuming it denests into a sum of two surds:

:sqrt{a+b sqrt{c}} = sqrt{d}+sqrt{e} :a+b sqrt{c} = d + e + 2 sqrt{de};

this can be solved by the quadratic formula.

In some cases, higher-power radicals may be needed to denest certain classes of nested radicals.

= See also =

Square_root#Infinitely_nested_square_roots

= References =

Decreasing the Nesting Depth of Expressions Involving Square Roots - http://www.almaden.ibm.com/cs/people/fagin/symb85.pdf

Simplifying Square Roots of Square Roots by Denesting - http://www.cybertester.com/data/denest.pdf

= External links =

Denesting nested radical

http://mathworld.wolfram.com/SquareRoot.html

Infinitely nested radical

http://mathworld.wolfram.com/NestedRadical.html