1. **Problem:** Simplify the expression $$\frac{1}{35} x n^{9n - 12 \times 27^n + 1}$$. 2. **Formula and rules:** - Multiplication and division of powers with the same base: $$a^m \times a^n = a^{m+n}$$ and $$\frac{a^m}{a^n} = a^{m-n}$$. - Simplify exponents carefully. 3. **Step-by-step simplification:** 1. The expression is $$\frac{1}{35} x n^{9n - 12 \times 27^n + 1}$$. 2. Note that $$12 \times 27^n$$ means $$12 \times (27^n)$$. 3. The exponent on $$n$$ is $$9n - 12 \times 27^n + 1$$, which cannot be simplified further without values for $$n$$. 4. The expression is already simplified as much as possible symbolically: $$\boxed{\frac{1}{35} x n^{9n - 12 \times 27^n + 1}}$$. **Final answer:** $$\frac{1}{35} x n^{9n - 12 \times 27^n + 1}$$. --- **Problem 2(b):** 1. A man invested 20000 in bank X and 25000 in bank Y. 2. Bank X pays simple interest at rate $$K\%$$ per annum. 3. Bank Y pays simple interest at rate $$1.5K\%$$ per annum. 4. Total interest from both banks after one year is 4600. 5. **Formula for simple interest:** $$I = P \times r \times t$$ where $$P$$ is principal, $$r$$ is rate (in decimal), $$t$$ is time in years. 6. Interest from bank X: $$I_X = 20000 \times \frac{K}{100} \times 1 = 200K$$. 7. Interest from bank Y: $$I_Y = 25000 \times \frac{1.5K}{100} \times 1 = 375K$$. 8. Total interest: $$I_X + I_Y = 200K + 375K = 575K$$. 9. Given total interest is 4600, so: $$575K = 4600$$ 10. Solve for $$K$$: $$K = \frac{4600}{575} = 8$$. **Final answer:** $$K = 8\%$$. --- **Summary:** - Simplified expression: $$\frac{1}{35} x n^{9n - 12 \times 27^n + 1}$$. - Interest rate $$K = 8\%$$.