We present a unified constructive digit-by-digit framework for exact root extraction using only integer arithmetic. The core contribution is a complete correctness theory for the fractional square root algorithm, proving that each computed decimal digit is exact and final, together with a sharp truncation error bound of $10^{-k}$ after $k$ digits. We further develop an invariant-based framework for computing the integer $e$-th root $\lfloor N^{1/e} \rfloor$ of a non-negative integer $N$ for arbitrary fixed exponents $e \ge 2$, derived directly from the binomial theorem. This method generalizes the classical long-division square root algorithm, preserves a constructive remainder invariant throughout the computation, and provides an exact decision procedure for perfect $e$-th power detection. We also explain why exact digit-by-digit fractional extraction with non-revisable digits is structurally possible only for square roots ($e=2$), whereas higher-order roots ($e \ge 3$) exhibit nonlinear coupling that prevents digit stability under scaling. All proofs are carried out in a constructive, algorithmic manner consistent with Bishop-style constructive mathematics, yielding explicit algorithmic witnesses, decidable predicates, and guaranteed termination. The resulting algorithms require no division or floating-point operations and are well suited to symbolic computation, verified exact arithmetic, educational exposition, and digital hardware implementation.

翻译：本文提出了一种统一的构造性逐位框架，用于仅使用整数运算的精确方根提取。核心贡献是分数平方根算法的完整正确性理论，证明了每个计算出的十进制数位都是精确且不可更改的，同时给出了计算$k$位后$10^{-k}$的严格截断误差界。我们进一步开发了基于不变量的框架，用于计算任意固定指数$e \ge 2$下非负整数$N$的整数$e$次方根$\lfloor N^{1/e} \rfloor$，该框架直接源自二项式定理。此方法推广了经典的长除法平方根算法，在整个计算过程中保持构造性余数不变量，并为完全$e$次幂检测提供了精确判定程序。我们还解释了为何具有不可修正数位的精确逐位分数提取在结构上仅适用于平方根（$e=2$），而高阶方根（$e \ge 3$）表现出非线性耦合，阻碍了缩放下的数位稳定性。所有证明均以符合Bishop风格构造性数学的构造性算法方式完成，产生显式算法见证、可判定谓词和保证终止性。所得算法无需除法或浮点运算，非常适用于符号计算、验证性精确算术、教学阐述和数字硬件实现。