which arises in discussing the regular pentagon, and more complicated ones such as
Rewriting a nested radical in this way is called denesting.
In the case of two nested square roots, the following theorem completely solves the problem of denesting.
However, Galois theory implies that either the left-hand side belongs to
This shows that the apparently more general denesting can always be reduced to the above one.
and, in the case of a minus in the right-hand side, (square roots are nonnegative by definition of the notation).
It follows by Vieta's formulas that x and y must be roots of the quadratic equation
For explicitly choosing the various signs, one must consider only positive real square roots, and thus assuming c > 0.
Thus, if the nested radical is real, and if denesting is possible, then a > 0.
Srinivasa Ramanujan demonstrated a number of curious identities involving nested radicals.
and In 1989 Susan Landau introduced the first algorithm for deciding which nested radicals can be denested.
Landau's algorithm involves complex roots of unity and runs in exponential time with respect to the depth of the nested radical.
[6] In trigonometry, the sines and cosines of many angles can be expressed in terms of nested radicals.
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
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.
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
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).
For n = 1, this root is the golden ratio φ, approximately equal to 1.618.
For example, it has been shown that nested square roots of 2 as[7]
These results can be used to obtain some nested square roots representations of
Ramanujan posed the following problem to the Journal of Indian Mathematical Society:
For a complete proof, we would need to show that this is indeed the solution to the equation for
Ramanujan stated the following infinite radical denesting in his lost notebook:
Viète's formula for π, the ratio of a circle's circumference to its diameter, is
In certain cases, infinitely nested cube roots such as
Again, by realizing that the whole expression appears inside itself, we are left with the equation
For n = 1, this root is the plastic ratio ρ, approximately equal to 1.3247.