Simplifying with the Exponent Rules

Simplifying with the exponent rules means applying the product, quotient, power, zero, negative, and power-of-a-product rules until each base appears once, with a positive exponent.

Simplify (2a³b²)² · a⁻⁴. The power of a product rule squares each factor: 2²a⁶b⁴ = 4a⁶b⁴. The product rule absorbs a⁻⁴: a⁶ · a⁻⁴ = a². The result is 4a²b⁴ — each base once, every exponent positive. The order of moves is flexible; the finish line is fixed: no repeated bases, no negative exponents, no zero exponents left unevaluated. A second run: simplify x⁵ · x⁰ / x³. The zero exponent rule collapses x⁰ to 1, and the quotient rule finishes: x⁵ / x³ = x². Two rules, two moves, done — and every path through the rules lands on the same simplified form.