Sure, I can help solve these expressions. Let's go through them step-by-step.

1) \(\frac{3}{4} \times \frac{5}{6} + \frac{2}{3}\)
- \(\frac{15}{24} + \frac{2}{3} = \frac{15}{24} + \frac{16}{24} = \frac{31}{24}\)

2) \(\left(-\frac{2}{7} + \frac{13}{8}\right) + \left(\frac{3}{8} + \frac{2}{7}\right)\)
- \(\left(-\frac{16}{56} + \frac{91}{56}\right) + \left(\frac{21}{56} + \frac{16}{56}\right)\)
- \(\frac{75}{56} + \frac{37}{56} = \frac{112}{56} = 2\)

3) \(\frac{11}{24} + \frac{25}{31} + \frac{13}{24} + 0.5 \times \frac{37}{31}\)
- \(\frac{24}{24} + \frac{25}{31} + \frac{6.5}{31}\)
- \(1 + \frac{31.5}{31} = \frac{62.5}{31} = 2.01613\)

4) \(\frac{11}{24} + \frac{5}{41} + \frac{13}{24} + 0.5 \times \frac{36}{41}\)
- \(\frac{24}{24} + \frac{5}{41} + \frac{6.5}{41}\)
- \(1 + \frac{11.5}{41} = \frac{52.5}{41} = 1.28049\)

5) \(\frac{-15}{11} + \frac{10}{12} \times \frac{15}{34} + \frac{4}{11} + \frac{18}{33}\)
- \(\frac{-15}{11} + \frac{150}{408} + \frac{4}{11} + \frac{6}{11}\)
- \(\frac{-15}{11} + \frac{25}{68} + \frac{4}{11} + \frac{6}{11}\)
- \(\frac{-7}{11} + \frac{25}{68}\)
- \(\approx -0.63636 + 0.36765 = -0.26871\)

6) \(\frac{15}{34} - \frac{1}{27} + \frac{19}{34} + \frac{20}{27} + \frac{21}{34} - \frac{34}{27}\)
- \(\frac{55}{34} + \frac{19}{27} - \frac{34}{27}\)
- \(\frac{55}{34} - \frac{15}{27}\)
- \(\approx 1.61765 - 0.55556 = 1.06209\)

7) \(\frac{1}{23} - \frac{1}{4} \times \frac{5}{13} = \frac{1}{5} \times \frac{1}{7}\)
- \(\frac{1}{23} - \frac{5}{52} = \frac{1}{35}\)
- \(-0.09686 + 0.02002 = -0.07684\)

8) \(\frac{9}{100} - \frac{23}{100} + \frac{9}{100} + \frac{81}{100} = \frac{9}{100} \times \frac{4}{\ 101}\)
- \(-\frac{23}{100} + \frac{9}{100} + \frac{81}{100} = \frac{4}{101}\)
- \(-0.14 + 0.09 + 0.81 = 0.76\)
- \(0.76 \neq 0.03960\)

9) \(\left(\frac{5}{4} - \frac{2}{3}\right) \times \frac{2022}{2023} = \left(\frac{3}{4} + \frac{3}{5}\right) \frac{2022}{2023}\)
- \(\left(0.5\right) \times \frac{2022}{2023} = \left(\frac{31}{20}\right) \frac{2022}{2023}\)
- \(\approx 0.9995\)

10) \(\frac{1}{4} - \frac{2}{5} \times \frac{3}{8} - \frac{3}{4} \times \frac{5}{3} \times \frac{3}{8}\)
- \(\frac{31}{20} - 1.40625 = -0.90625\)

11) \(\left(\frac{1}{3)} \times \frac{3}{10} \right) \div \left(\frac{3}{7\ } \times \frac{2}{3}\right) \times \frac{10}{3} \times \frac{3}{5}\)
- \(\approx 0.72917\)

12) \(\left(\frac{4}{15} \times \frac{5}{6}\right) \times \left(\frac{11}{15} \div \frac{5}{6}\right)\)
- \(\approx \frac{220}{225} = 0.97778\)

13) \(21 \times \frac{1}{5} - \left(\frac{5}{2}\right) - 16 \times \left(\frac{-5}{2}\right)\)
- \(4.2 - 2.5 + 40 = 41.7\)

14) \(\frac{11}{15} \left(\frac{-19}{13}\right) + \frac{11}{15} \left(\frac{-7}{13}\right) \times \frac{4}{5}\)
- \(\approx -\frac{209}{195} - \frac{28}{15} = -2.71282\)

Let me know if you need further clarification!