Nested Radical - Denesting Nested Radicals

Denesting Nested Radicals

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 using the quadratic formula and 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

squaring both sides yields

By plugging in a-e for d one obtains

Rearranging terms will give an quadratic equation which can be solved for e

The solution d is the algebraic conjugate of e. If

then

However, this approach works for nested radicals of the form

if and only if

is an integer, in which case the nested radical can be denested into a sum of surds.

In some cases, higher-power radicals may be needed to denest the nested radical.