1. **State the problem:** Simplify the expression $\left(2x^{2}y^{-2}m^{3}\right)^{4} \times \left(2x^{4}y^{5}m^{8}\right)$ and find which option (A, B, C, or D) matches the result. 2. **Recall the exponent rules:** - Power of a product: $\left(ab\right)^n = a^n b^n$ - Power of a power: $\left(x^a\right)^b = x^{ab}$ - Multiply like bases: $x^a \times x^b = x^{a+b}$ 3. **Apply the power to each factor inside the first parentheses:** $$\left(2x^{2}y^{-2}m^{3}\right)^4 = 2^4 \times x^{2 \times 4} \times y^{-2 \times 4} \times m^{3 \times 4} = 16x^{8}y^{-8}m^{12}$$ 4. **Multiply this result by the second expression:** $$16x^{8}y^{-8}m^{12} \times 2x^{4}y^{5}m^{8}$$ 5. **Multiply coefficients and add exponents of like bases:** - Coefficients: $16 \times 2 = 32$ - $x$ powers: $8 + 4 = 12$ - $y$ powers: $-8 + 5 = -3$ - $m$ powers: $12 + 8 = 20$ 6. **Final simplified expression:** $$32x^{12}y^{-3}m^{20}$$ 7. **Compare with options:** - A: $32x^{10}y^{7}m^{15}$ (No) - B: $32x^{12}y^{-3}m^{20}$ (Yes) - C: $16x^{10}y^{7}m^{15}$ (No) - D: $16x^{12}y^{-3}m^{20}$ (No) **Answer:** Option B is correct.