Geometric Construction for Negative Rational Exponents

This GeoGebra example visually demonstrates how negative rational exponents connect to the reciprocal of a corresponding positive rational power (e.g., a^(-p/q) as the inverse of a^(p/q)). It highlights key ideas: rational exponent rules, the equivalence between rational powers and roots, and the interpretation of negative exponents as reciprocals. Construction-wise, the diagram is built from two related sets of labeled points (A2, A3, A1; B1–H1; and B2–H2) and their connecting segments/figures. By positioning these points to enforce proportional relationships and reuse the same base geometry at two levels (the “positive power” layer and the “negative power” layer), the construction turns algebraic exponent manipulation into geometric ratio transformations. Suitable for secondary advanced algebra and early college math instruction, especially when teaching exponent/roots laws and using visual models to reason about reciprocal relationships.