1. **State the problem:** Simplify the expression $x^2 y^4 (x^8 y^5)^4$. 2. **Recall the exponent rules:** - When raising a power to another power, multiply the exponents: $(a^m)^n = a^{m \times n}$. - When multiplying like bases, add the exponents: $a^m \times a^n = a^{m+n}$. 3. **Apply the power to the parentheses:** $$ (x^8 y^5)^4 = x^{8 \times 4} y^{5 \times 4} = x^{32} y^{20} $$ 4. **Multiply the original terms:** $$ x^2 y^4 \times x^{32} y^{20} = x^{2+32} y^{4+20} = x^{34} y^{24} $$ 5. **Final simplified expression:** $$ x^{34} y^{24} $$ 6. **Answer choice:** $x^{34} y^{24}$ is the correct simplification.