1. Stating the problem: Simplify the expressions $2x y^2 \times 3x^2 y^2$ and $3y^2 \times 4x d y^3$. 2. For multiplication of terms with the same base, use the rule $a^m \times a^n = a^{m+n}$. 3. Simplify the first expression: $$2x y^2 \times 3x^2 y^2 = (2 \times 3)(x^{1} \times x^{2})(y^{2} \times y^{2})$$ $$= 6 x^{1+2} y^{2+2} = 6 x^{3} y^{4}$$ 4. Simplify the second expression: $$3 y^{2} \times 4 x d y^{3} = (3 \times 4) (y^{2} \times y^{3}) x d = 12 x d y^{2+3} = 12 x d y^{5}$$ Final answers: - First expression: $6 x^{3} y^{4}$ - Second expression: $12 x d y^{5}$