Convert Between Exponential and Radical Forms - Algebra-2

1. Fundamental Concepts

Composition of Forms
- Radical Form: , consisting of the radical symbol, index n (default ), and radicand a (for even roots, ; for odd roots, a is any real number).
- Exponential Form: , consisting of the base a (corresponding to the radicand in radical form) and exponent b (convertible when b is a rational number).
Bridge for Conversion: Rational exponent (where m and n are integers, ), connecting the operations of "taking roots" and "raising to powers".

Core Rules
- Positive Rational Exponent: (the denominator n is the index of the root, and the numerator m is the power).
- Negative Rational Exponent: (where ; first take the reciprocal, then convert to radical form).
Special Cases
- When : (e.g., ).
- When : (the index is omitted, e.g., ).
- Zero Exponent: (where ), corresponding to .

Conversion with negative exponent: .
Practical Application: The potential energy formula is converted to radical form as . Substituting and , we get .

Label Parameters: Before conversion, mark a (base/radicand), m (numerator of the exponent), and n (index of the root) to avoid confusion.
Simplify the Base: Decompose the base into a power form (e.g., ) before conversion (e.g., ).
Two-Step for Negative Exponents: First take the reciprocal to eliminate the negative sign, then convert to radical form (e.g., ).
Unify Forms: For mixed forms, first unify them into rational exponents or radical forms (e.g., ).