1. **State the problem:** Evaluate the expression $$\left(\frac{1}{3} + \frac{2}{9}\right)^9 + \frac{\left(\frac{3}{4}\right)^2 / 9}{\frac{9}{4} \times (-4)}$$ and determine which of the options A) 4, B) 6, or C) 9 is correct. 2. **Simplify inside the parentheses:** $$\frac{1}{3} + \frac{2}{9} = \frac{3}{9} + \frac{2}{9} = \frac{5}{9}$$ 3. **Calculate the first term:** $$\left(\frac{5}{9}\right)^9$$ 4. **Simplify the numerator of the second term:** $$\left(\frac{3}{4}\right)^2 = \frac{9}{16}$$ 5. **Divide by 9:** $$\frac{9}{16} \div 9 = \frac{9}{16} \times \frac{1}{9} = \frac{1}{16}$$ 6. **Simplify the denominator of the second term:** $$\frac{9}{4} \times (-4) = \frac{9}{4} \times \left(-4\right) = -9$$ 7. **Calculate the second term:** $$\frac{\frac{1}{16}}{-9} = \frac{1}{16} \times \left(-\frac{1}{9}\right) = -\frac{1}{144}$$ 8. **Combine both terms:** $$\left(\frac{5}{9}\right)^9 - \frac{1}{144}$$ 9. **Estimate $$\left(\frac{5}{9}\right)^9$$:** Since $$\frac{5}{9} \approx 0.5556$$, raising to the 9th power makes it very small but positive. The exact value is approximately $$0.00195$$. 10. **Sum the terms:** $$0.00195 - 0.00694 = -0.00499$$ (approximately) 11. **Interpretation:** The result is approximately $$-0.005$$, which is closest to none of the positive options 4, 6, or 9. 12. **Re-examine the problem:** The problem likely expects the value of the first term only or a different interpretation. Since the second term is negative and small, the first term dominates. 13. **Check if the problem expects the first term only:** $$\left(\frac{5}{9}\right)^9 \approx 0.00195$$ which is much less than 4, 6, or 9. 14. **Check if the problem expects the sum of absolute values:** $$0.00195 + 0.00694 = 0.00889$$ still less than all options. 15. **Conclusion:** The expression evaluates to approximately $$-0.005$$, which does not match any of the options A, B, or C. **Final answer:** None of the options A) 4, B) 6, or C) 9 matches the evaluated expression. If the problem expects the value of $$\left(\frac{1}{3} + \frac{2}{9}\right)^9$$ alone, the answer is approximately 0.00195. If the problem expects the value of $$\frac{\left(\frac{3}{4}\right)^2 / 9}{\frac{9}{4}(-4)}$$ alone, the answer is $$-\frac{1}{144}$$. Since none of these match the options, please verify the problem statement or options.