An Analytical Study of the Nested Radical Identity for sqrt(a ± b) and Its Applications in Algebraic Simplification

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Scientific Essay from the year 2026 in the subject Mathematics - General, Basics, grade: C, , language: English, abstract: This article examines the nested radical identity sqrt(a ± b) = sqrt( (a + sqrt(a^2 - b^2)) / 2 ) ± sqrt( (a - sqrt(a^2 - b^2)) / 2 ). We analyze its mathematical foundation, conditions of validity, and applications in simplifying complex radical expressions. The identity proves useful in algebra, number theory, and computational mathematics for denesting radicals and enhancing computational efficiency. Numerical verification through Python code confirms exact equivalence within floating-point precision.

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