of the decomposition over and 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 (≠0, otherwise would be the square of ), hence and must be and Thus and are rational if and only if is a rational number. For explicitly choosing the various signs, one must consider only positive real square roots, and thus assuming . The equation 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: and 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