\(\left(\frac{16}{81}\right)^{-\frac{3}{4}}\times \sqrt{\frac{100}{81}}\) = \(\frac{1}{\left(\sqrt[4]{\frac{16}{81}}\right)^3}\times \frac{10}{9}\) = \(\frac{1}{\left(\frac{2}{3}\right)^3}\times\frac{10}{9}\) = \(\frac{27}{8}\times \frac{10}{9} = \frac{15}{4}\) OR We start with the expression: \(\left(\frac{16}{81}\right)^{-\frac{3}{4}} \times \sqrt{\frac{100}{81}}\) First, simplify \(\left(\frac{16}{81}\right)^{-\frac{3}{4}}\): \(\left(\frac{16}{81}\right)^{-\frac{3}{4}} = \left(\frac{81}{16}\right)^{\frac{3}{4}} = \frac{81^{\frac{3}{4}}}{16^{\frac{3}{4}}}\) Calculating \(81^{\frac{3}{4}}\) and \(16^{\frac{3}{4}}\): \(81 = 3^4 \implies 81^{\frac{3}{4}} = 3^3\) = 27 \(16 = 4^2 \implies 16^{\frac{3}{4}} = 4^{\frac{3}{2}} = 2^3\) = 8 Thus, \(\left(\frac{16}{81}\right)^{-\frac{3}{4}} = \frac{27}{8}\) Next, simplify \(\sqrt{\frac{100}{81}}\): \(\sqrt{\frac{100}{81}} = \frac{\sqrt{100}}{\sqrt{81}} = \frac{10}{9}\) Now, multiply the two results: \(\frac{27}{8} \times \frac{10}{9} = \frac{27 \times 10}{8 \times 9} = \frac{270}{72}\) Finding the GCD of 270 and 72 gives: \(\frac{270 \div 18}{72 \div 18} = \frac{15}{4}\) The final result is: \(\left(\frac{16}{81}\right)^{-\frac{3}{4}} \times \sqrt{\frac{100}{81}} = \frac{15}{4}\)