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(9/16)^2016.(16/9)^2015.4/3
9/16.(9/16)^2015.(16/9)^2015.4/3
9/16.(9/16.16/9)^2015.4/3
9/16.1^2015.4/3
9/16.1.4/3
=>9/16.4/3=3/4
\(\frac{3}{4}\cdot\frac{8}{9}\cdot\frac{15}{16}\cdot...\cdot\frac{9999}{10000}\)
\(=\frac{1\cdot3}{2^2}\cdot\frac{2\cdot4}{3^2}\cdot\frac{3\cdot5}{4^2}\cdot...\cdot\frac{99\cdot101}{100^2}\)
\(=\frac{\left(1\cdot3\right)\left(2\cdot4\right)\left(3\cdot5\right)...\left(99\cdot101\right)}{2^2\cdot3^2\cdot4^2\cdot...\cdot100^2}\)
\(=\frac{\left(1\cdot2\cdot3\cdot...\cdot99\right)\left(3\cdot4\cdot5\cdot101\right)}{\left(2\cdot3\cdot4\cdot...\cdot100\right)\left(2\cdot3\cdot4\cdot...\cdot100\right)}\)
\(=2\cdot101=202\)
\(\frac{\frac{2}{3}+\frac{2}{5}+\frac{2}{7}+\frac{2}{9}}{\frac{11}{3}+\frac{11}{5}+\frac{11}{7}+\frac{11}{9}}=\frac{2\left(\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{9}\right)}{11\left(\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{9}\right)}=\frac{2}{11}\)
\(\left(\frac{9}{16}\right)^{2016}.\left(\frac{16}{9}\right)^{2015}.\frac{4}{3}\)
\(=\left(\frac{9}{16}\right)^{2015}.\left(\frac{16}{9}\right)^{2015}.\frac{9}{16}.\frac{4}{3}\)
\(=\left(\frac{9}{16}.\frac{16}{9}\right)^{2015}.\frac{3}{4}\)
\(=\frac{1.3}{4}=\frac{3}{4}\)