We can evaluate the expression \(\frac{1}{3} \div [\frac{5}{7}(\frac{9}{10} -1 + \frac{3}{4})]\) by first simplifying the expression inside the square brackets, and then dividing \(\frac{1}{3}\) by the result. Let's start by simplifying the expression inside the square brackets: \(\frac{5}{7}(\frac{9}{10} -1 + \frac{3}{4})\) = \(\frac{5}{7}(\frac{9}{10} - \frac{10}{10} + \frac{3}{4})\) = \(\frac{5}{7}(\frac{-1}{10} + \frac{3}{4})\) = \(\frac{5}{7}(\frac{-2}{20} + \frac{15}{20})\) = \(\frac{5}{7}(\frac{13}{20})\) = \(\frac{65}{140}\) Now we can divide \(\frac{1}{3}\) by this result: \(\frac{1}{3} \div [\frac{65}{140}]\) = \(\frac{1}{3} [\frac{140}{65}]\) = \(\frac{140}{195}\) = \(\frac{28}{39}\) So, the value of the expression is \(\frac{28}{39}\).