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:
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
However, this approach works for nested radicals of the form if and only if is a rational number, 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.
In the case in which the cubic has only one real root, the real root is given by this expression with the radicands of the cube roots being real and with the cube roots being the real cube roots. In the case of three real roots, the square root expression is an imaginary number; here any real root is expressed by defining the first cube root to be any specific complex cube root of the complex radicand, and by defining the second cube root to be the complex conjugate of the first one. The nested radicals in this solution cannot in general be simplified unless the cubic equation has at least one rational solution. Indeed, if the cubic has three irrational but real solutions, we have the casus irreducibilis, in which all three real solutions are written in terms of cube roots of complex numbers. On the other hand, consider the equation
which has the rational solutions 1, 2, and —3. The general solution formula given above gives the solutions
For any given choice of cube root and its conjugate, this contains nested radicals involving complex numbers, yet it is reducible (even though not obviously so) to one of the solutions 1, 2, or –3.
Under certain conditions infinitely nested square roots such as
represent rational numbers. This rational number can be found by realizing that x also appears under the radical sign, which gives the equation
If we solve this equation, we find that x = 2 (the second solution x = −1 doesn't apply, under the convention that the positive square root is meant). This approach can also be used to show that generally, if n > 0, then
and is the positive root of the equation x2 − x − n = 0. For n = 1, this root is the golden ratio φ, approximately equal to 1.618. The same procedure also works to obtain
which is the positive root of the equation x2 + x − n = 0.
The value obtained for the infinite nested radical by converting to a polynomial equation and solving is valid only if the sequence of values, obtained by successively nesting more and more radicals, converges. For example, the above-considered expression
if convergent, is the limiting value of the process
starting from the initial value We have
Convergence requires that the absolute value of this expression be less than 1 in the neighborhood of the value of x given earlier that satisfies the corresponding polynomial equation. It turns out that this condition is that n > 3/4, which holds if, for example, we require the positive number n to be an integer. Then it is sufficient for convergence that the initial value be in the basin of attraction of the indicated stationary value of x.