Denesting radicals

In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include

which arises in discussing the regular pentagon, and more complicated ones such as

Some nested radicals can be rewritten in a form that is not nested. For example,

Rewriting a nested radical in this way is called denesting. This process is generally considered a difficult problem, although a special class of nested radical can be denested by assuming it denests into a sum of two surds:

Squaring both sides of this equation yields:

This can be solved by finding two numbers such that their sum is equal to a and their product is b²c/4, or by equating coefficients of like terms—setting rational and irrational parts on both sides of the equation equal to each other. The solutions for e and d can be obtained by first equating the rational parts:

which gives

For the irrational parts note that

and squaring both sides yields

By plugging in a − d for e one obtains

Rearranging terms will give a quadratic equation which can be solved for d using the quadratic formula:

Since a = d+e, the solution e is the algebraic conjugate of d. If we set

then

However, this approach works for nested radicals of the form ${\displaystyle {\sqrt {a\pm b{\sqrt {c}}\ }}}$ if and only if ${\displaystyle {\sqrt {a^{2}-b^{2}c}}}$ is a rational number, in which case the nested radical can be denested into a sum of surds.

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