Nested radical

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
Denesting.
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 is not always possible, and, even when possible, it is often difficult.
Two nested square roots.
In the case of two nested square roots, the following theorem completely solves the problem of denesting.
If and are rational numbers and is not the square of a rational number, there are two rational numbers and such that
if and only if formula_7 is the square of a rational number .
If the nested radical is real, and are the two numbers
In particular, if and are integers, then and are integers.
This result includes denestings of the form
as may always be written formula_12 and at least one of the terms must be positive (because the left-hand side of the equation is positive).
A more general denesting formula could have the form
However, Galois theory implies that either the left-hand side belongs to formula_14 or it must be obtained by changing the sign of either formula_15 formula_16 or both. In the first case, this means that one can take and formula_17 In the second case, formula_18 and another coefficient must be zero. If formula_19 one may rename as for getting formula_20 Proceeding similarly if formula_21 it results that one can suppose formula_22 This shows that the apparently more general denesting can always be reduced to the above one.
"Proof": By squaring, the equation 
is equivalent with 
and, in the case of a minus in the right-hand side, 
(square roots are nonnegative by definition of the notation). As the inequality may always be satisfied by possibly exchanging and , solving the first equation in and is equivalent with solving 
This equality implies that formula_26 belongs to the quadratic field formula_27 In this field every element may be uniquely written formula_28 with formula_18 and formula_30 being rational numbers. This implies that formula_31 is not rational (otherwise the right-hand side of the equation would be rational; but the left-hand side is irrational). As and must be rational, the square of formula_31 must be rational. This implies that formula_33 in the expression of formula_31 as formula_35 Thus
for some rational number formula_37
The uniqueness of the decomposition over and formula_38 implies thus that the considered equation is equivalent with
It follows by Vieta's formulas that and must be roots of the quadratic equation
its formula_41 (≠0, otherwise would be the square of ), hence and must be
Thus and are rational if and only if formula_44 is a rational number.
For explicitly choosing the various signs, one must consider only positive real square roots, and thus assuming . The equation formula_45 shows that . Thus, if the nested radical is real, and if denesting is possible, then . Then, the solution writes
Some identities of Ramanujan.
Srinivasa Ramanujan demonstrated a number of curious identities involving nested radicals. Among them are the following:
Other odd-looking radicals inspired by Ramanujan include:
Landau's algorithm.
In 1989 Susan Landau introduced the first algorithm for deciding which nested radicals can be denested. Earlier algorithms worked in some cases but not others. Landau's algorithm involves complex roots of unity and runs in exponential time with respect to the depth of the nested radical.
In trigonometry.
In trigonometry, the sines and cosines of many angles can be expressed in terms of nested radicals. For example,
and
The last equality results directly from the results of .
In the solution of the cubic equation.
Nested radicals appear in the algebraic solution of the cubic equation. Any cubic equation can be written in simplified form without a quadratic term, as
whose general solution for one of the roots is
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.
Infinitely nested radicals.
Square roots.
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 "x"2 − "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, if "n" > 1,
which is the positive root of the equation "x"2 + "x" − "n" = 0.
Ramanujan's infinite radicals.
Ramanujan posed the following problem to the "Journal of Indian Mathematical Society":
This can be solved by noting a more general formulation:
Setting this to "F"("x") and squaring both sides gives us
which can be simplified to
It can then be shown that
So, setting "a" = 0, "n" = 1, and "x" = 2, we have
Ramanujan stated the following infinite radical denesting in his lost notebook:
The repeating pattern of the signs is formula_70
Viète's expression for.
Viète's formula for , the ratio of a circle's circumference to its diameter, is
Cube roots.
In certain cases, infinitely nested cube roots such as
can represent rational numbers as well. Again, by realizing that the whole expression appears inside itself, we are left with the equation
If we solve this equation, we find that "x" = 2. More generally, we find that
is the positive real root of the equation "x"3 − "x" − "n" = 0 for all "n" > 0. For "n" = 1, this root is the plastic number "ρ", approximately equal to 1.3247.
The same procedure also works to get
as the real root of the equation "x"3 + "x" − "n" = 0 for all "n" > 1.
Herschfeld's Convergence Theorem.
An infinitely nested radical formula_76 (where all formula_77 are nonnegative) converges if and only if there is some formula_78 such that formula_79 for all formula_80. 
Proof of "if".
We observe that 
Moreover, the sequence formula_82 is monotonically increasing. Therefore it converges, by the monotone convergence theorem.
Proof of "only if".
If the sequence formula_82 converges, then it is bounded.
However, formula_84, hence formula_85 is also bounded.