1. **State the problem:** Simplify the expression $$\frac{(10m^5 n^7)^3}{(5mn^3)^2}$$ and determine which multiple-choice answer is correct. 2. **Write the formula and rules:** When raising a power to another power, multiply the exponents: $$(a^m)^n = a^{mn}$$ When dividing like bases, subtract exponents: $$\frac{a^m}{a^n} = a^{m-n}$$ When multiplying coefficients, multiply normally. 3. **Apply the power to each factor in numerator:** $$ (10m^5 n^7)^3 = 10^3 \cdot (m^5)^3 \cdot (n^7)^3 = 1000 m^{15} n^{21} $$ 4. **Apply the power to each factor in denominator:** $$ (5mn^3)^2 = 5^2 \cdot m^2 \cdot (n^3)^2 = 25 m^2 n^6 $$ 5. **Write the full fraction:** $$ \frac{1000 m^{15} n^{21}}{25 m^2 n^6} $$ 6. **Simplify coefficients:** $$ \frac{1000}{25} = \cancel{\frac{1000}{25}} = 40 $$ 7. **Simplify variables by subtracting exponents:** $$ m^{15-2} = m^{13} $$ $$ n^{21-6} = n^{15} $$ 8. **Final simplified expression:** $$ 40 m^{13} n^{15} $$ 9. **Answer:** The correct choice is **40m^13 n^15**.