1. **State the problem:** Simplify the expression $$\frac{4(x^2 y)^3 \times 2xy^2}{2x^2 y}$$. 2. **Recall the exponent and multiplication rules:** - When raising a power to another power, multiply the exponents: $$(a^m)^n = a^{mn}$$. - When multiplying like bases, add the exponents: $$a^m \times a^n = a^{m+n}$$. - When dividing like bases, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. 3. **Apply the power to the term inside the parentheses:** $$ (x^2 y)^3 = x^{2 \times 3} y^3 = x^6 y^3 $$ 4. **Rewrite the numerator:** $$ 4(x^2 y)^3 \times 2xy^2 = 4 \times x^6 y^3 \times 2 \times x y^2 = 8 x^{6+1} y^{3+2} = 8 x^7 y^5 $$ 5. **Rewrite the denominator:** $$ 2 x^2 y $$ 6. **Form the fraction:** $$ \frac{8 x^7 y^5}{2 x^2 y} $$ 7. **Simplify the fraction by dividing coefficients and subtracting exponents:** $$ \frac{8}{2} = \cancel{\frac{8}{2}} = 4 $$ $$ x^{7-2} = x^{\cancel{7-2}} = x^5 $$ $$ y^{5-1} = y^{\cancel{5-1}} = y^4 $$ 8. **Final simplified expression:** $$ 4 x^5 y^4 $$ **Answer:** $$4 x^5 y^4$$