Practice with Rational Bases via Geometric Mean Construction

This GeoGebra example demonstrates how powers with rational bases can be connected to the geometric mean, allowing students to verify algebraic relationships through a constructed plane figure. The key math ideas include geometric mean (Geometric Mean), rational-exponent/power behavior with rational bases (Rational Bases), and the interpretation of numerical relationships as measurable geometric quantities. Construction-wise, the setup relies on interactive inputs (InputBox1 through InputBox9) and on displayed/updated labels via Textfield. It also creates and uses core objects such as points A and B and multiple function/derived objects f1 through f9 to represent the relevant power/measurement values on the diagram, highlighting how the geometry supports the expected equality/inequality structure. Suitable for middle school to early high school topics covering indices, radicals, and geometric-algebra connections, as well as classroom exploration or guided practice.