Computational mastery in algebra demands fluency with repeated operations, structural powers, and roots. Exponents provide a compact mathematical notation for repeated multiplication, governed by universal index laws: the product of powers rule, the quotient rule, the power of a power theorem, and zero or negative exponent definitions (a^0 = 1, a^-n = 1/a^n). Complementing exponential growth, radicals reverse these operations through square roots and cube roots in Q, demonstrating that while the square root of a negative rational is undefined in Q, the cube root of a negative rational is always a well-defined negative rational. Meanwhile, the absolute value function |x| measures pure geometric distance from zero on the number line. In this rigorous lesson, students develop fluency across powers, roots, and absolute value equations.