1. Problem: Simplify $ (4n^7)^5$.\n 2. Formula and rules: Use the power-of-a-product rule $ (ab)^m = a^m b^m$ and the power-of-a-power rule $ (a^m)^n = a^{mn}$.\n 3. Apply the power-of-a-product rule to separate the coefficient and the power of n.\n $$ (4n^7)^5 = 4^5 (n^7)^5 $$\n 4. Apply the power-of-a-power rule to the n-term and compute the exponent.\n $$ (n^7)^5 = n^{7\cdot5} = n^{35} $$\n 5. Compute the power of the coefficient.\n $$ 4^5 = 1024 $$\n 6. Combine the results to get the simplified expression.\n $$ (4n^7)^5 = 1024\,n^{35} $$\n 7. Final answer: $1024\,n^{35}$.\n 8. Note: None of the provided multiple-choice options match this correct result.\n