6) \(\frac{8}{25}\cdot\frac{45}{12}:\left(1\frac{1}{14}-1\frac{1}{21}\right)\cdot\frac{5}{27}+5\frac{2}{3}\)

Сначала вычислим скобки:

\(1\frac{1}{14}-1\frac{1}{21}=\frac{15}{14}-\frac{22}{21}=\frac{45-44}{42}=\frac{1}{42}\).

\(\frac{8}{25}\cdot\frac{45}{12}=\frac{6}{5}\).

\(\frac{6}{5}:\frac{1}{42}\cdot\frac{5}{27}=\frac{6}{5}\cdot42\cdot\frac{5}{27}=\frac{28}{3}\).

\(5\frac{2}{3}=\frac{17}{3}\), поэтому \(\frac{28}{3}+\frac{17}{3}=\frac{45}{3}=15\).

Ответ: \(15\).