Rational exponents

Connect fractional exponent notation with roots and powers, then evaluate expressions on their permitted real domains. This set covers the meaning of the numerator and denominator of a rational exponent, equivalent radical forms, positive and fractional bases, negative exponents and the difference between a negative base and a minus sign outside a power.

For a negative base, reduce the rational exponent to lowest terms before checking whether its denominator is odd or even. Questions also distinguish a positive power of zero from an undefined reciprocal, identify allowed variable values, and apply product, quotient and power rules under appropriate assumptions. A direct counterexample explains why a rule valid for positive bases cannot be extended indiscriminately to nested powers of negative numbers.

Single-answer and select-all tasks progress from translation and evaluation to domain analysis, several-step reasoning and applications. Geometric scaling and the comparison of powers with a base between zero and one build meaning beyond notation. Review integer exponents, principal roots and fraction arithmetic first. Explanations identify both the calculation and its conditions, giving a foundation for later work with power functions and equations without requiring logarithms or complex numbers.