Index laws reverse your intuition: x² × x³ isn't x⁶, it's x⁵. Powers combine by adding, dividing subtracts, and raising a power multiplies the exponents. Drag the slider to watch how changing the base transforms powers across different exponent rules.
When you multiply powers with the same base, you add the exponents: x² × x³ = x⁽²⁺³⁾ = x⁵. This isn't obvious until you expand it: (x·x) × (x·x·x) gives you five x's total. The operation on the bases (multiplication) becomes a different operation on the exponents (addition). This pattern holds for all real bases and exponents, making index laws the foundation for logarithms, exponential growth, and calculus differentiation rules.