1. **State the problem:** Evaluate the expression $$\left[\left(\frac{2}{3} + \frac{3}{4}\right)^2 \div \left(\frac{5}{6} - \frac{1}{3}\right)\right] + \sqrt{81} - 2^3$$. 2. **Recall formulas and rules:** - Addition and subtraction of fractions require a common denominator. - Division of fractions is multiplication by the reciprocal. - Square root of a perfect square is the positive root. - Exponentiation follows order of operations. 3. **Calculate inside the first parentheses:** $$\frac{2}{3} + \frac{3}{4} = \frac{8}{12} + \frac{9}{12} = \frac{17}{12}$$ 4. **Square the sum:** $$\left(\frac{17}{12}\right)^2 = \frac{289}{144}$$ 5. **Calculate the denominator inside the division:** $$\frac{5}{6} - \frac{1}{3} = \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$$ 6. **Divide the squared sum by the denominator:** $$\frac{289}{144} \div \frac{1}{2} = \frac{289}{144} \times 2 = \frac{578}{144} = \frac{289}{72}$$ 7. **Calculate the square root:** $$\sqrt{81} = 9$$ 8. **Calculate the power:** $$2^3 = 8$$ 9. **Combine all parts:** $$\frac{289}{72} + 9 - 8 = \frac{289}{72} + 1 = \frac{289}{72} + \frac{72}{72} = \frac{361}{72}$$ 10. **Simplify the fraction if possible:** 361 and 72 share no common factors other than 1, so the fraction is simplified. **Final answer:** $$\frac{361}{72}$$