Simplify: (3x²)(4x³)

Multiply coefficients and apply exponent rules to variables. Coefficients: 3 × 4 = 12. Variables: x² × x³ — use the product rule a^m × a^n = a^(m+n), so x² × x³ = x^(2+3) = x⁵. Combined: 12x⁵. Common error: multiplying exponents instead of adding (multiplication of like bases ADDS exponents; raising a power to a power MULTIPLIES exponents). Distinguish: (1) x² × x³ = x⁵ (add); (2) (x²)³ = x⁶ (multiply). Other key exponent operations: (1) (3x²y)(4xy³) = 12 x³y⁴ (multiply coefficients, add exponents of each variable separately); (2) 12x⁵ ÷ 3x² = 4x³ (divide coefficients, subtract exponents); (3) (2x³)² = 4x⁶ (raise both coefficient and variable to the outside exponent); (4) (3x²)³ = 27x⁶ (3³ = 27, and (x²)³ = x⁶). Master these patterns to handle polynomial arithmetic efficiently.