1. **State the problem:** Simplify the expression $$\frac{(5n^8p^6)^2}{30n^4p^2}$$. 2. **Apply the power to each factor inside the parentheses:** $$(5n^8p^6)^2 = 5^2 \cdot (n^8)^2 \cdot (p^6)^2 = 25n^{16}p^{12}$$ 3. **Rewrite the expression with the expanded numerator:** $$\frac{25n^{16}p^{12}}{30n^4p^2}$$ 4. **Divide the coefficients and apply the quotient rule for exponents:** $$\frac{25}{30} \cdot \frac{n^{16}}{n^4} \cdot \frac{p^{12}}{p^2}$$ 5. **Simplify the coefficient fraction:** $$\frac{\cancel{25}}{\cancel{30}} = \frac{5}{6}$$ 6. **Simplify the variables using the rule $\frac{a^m}{a^n} = a^{m-n}$:** $$n^{16-4} = n^{12}$$ $$p^{12-2} = p^{10}$$ 7. **Combine all simplified parts:** $$\frac{5}{6} n^{12} p^{10}$$ **Final answer:** $$\frac{5}{6} n^{12} p^{10}$$