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 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, if n > 1, which is the positive root of the equation x2 + x − n = 0. Nested square roots of 2 The nested square roots of 2 are a special case of the wide class of infinitely nested radicals. There are many known results that bind them to sines and cosines. For example, it has been shown that nested square roots of 2 as where with in [−2,2] and for , are such that for This result allows to deduce for any the value of the following infinitely nested radicals consisting of k nested roots as If , then These results can be used to obtain some nested square roots representations of . Let us consider the term defined above. Then where . 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, assuming is analytic, 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 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