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6)

$\big[\big(0,1\overline6+\frac{2}{3}\big)÷\big(\frac{4}{3}-2\big)\big]×\big[1,\overline3÷\big(0,2-\frac{2}{3}\big)\big]=$

$= \big[\big(\frac{16-1}{90}+\frac{2}{3}\big)÷\big(\frac{4-6}{3}\big)\big]×\big[\frac{13-1}{9}÷\big(\frac{2}{10}-\frac{2}{3}\big)\big]=$

$= \big[\big(\frac{15}{90}+\frac{2}{3}\big)÷\big(-\frac{2}{3}\big)\big]×\big[\frac{12}{9}÷\big(\frac{1}{5}-\frac{2}{3}\big)\big]=$

$= \big[\big(\frac{1}{6}+\frac{2}{3}\big)÷\big(-\frac{2}{3}\big)\big]×\big[\frac{4}{3}÷\big(\frac{3-10}{15}\big)\big]=$

$= \big[\big(\frac{1+4}{6}\big)×\big(-\frac{3}{2}\big)\big]×\big[\frac{4}{3}÷\big(-\frac{7}{15}\big)\big]=$

$= \big[\frac{5}{6}×\big(-\frac{3}{2}\big)\big]×\big[\frac{4}{3}×\big(-\frac{15}{7}\big)\big]=$

$= \big[\frac{5}{2}×\big(-\frac{1}{2}\big)\big]×\big[\frac{4}{1}×\big(-\frac{5}{7}\big)\big]=$

$= \big(-\frac{5}{4}\big)×\big(-\frac{20}{7}\big)=$

$= \big(-\frac{5}{1}\big)×\big(-\frac{5}{7}\big)=$

$=\frac{25}{7}$