Decidability of Equality of Radical Expressions
Algebraic numbers are solutions of polynomials with rational coefficients.
$+,\times,-,/$ of algebraic numbers result in algebraic numbers because algebraic numbers form a field (1). This means nested radicals are algebraic numbers too (2).
Nested radicals can be denested by algorithm (3,4).
Each algebraic number of degree $n$ can be uniquely represented as a $n$ by $n$ matrix of integers under a suitable basis (for example, $[1,x, (x^2+1)/2]$). This representation allows symbolic evaluation of $+,\times,-,/$ by matrix addition, multiplication, and inverse (p.159 of 5,6,7).
Two terms are equal if their unique representations are identical.