Tìm \[x\]:

a) \[\frac{{11}}{8} - \frac{3}{8} \cdot x = \frac{1}{8}\];                                          b) \({\left( {{\rm{x}} - \frac{1}{2}} \right)^{\rm{2}}}{\rm{ = }}\frac{{\rm{1}}}{4}\).

Ta có \[A = \left( {1 + \frac{1}{{1.3}}} \right).\left( {1 + \frac{1}{{2.4}}} \right).\left( {1 + \frac{1}{{3.5}}} \right).\,\,...\,\,.\left( {1 + \frac{1}{{2022.2024}}} \right)\,\]

\[ = \frac{4}{{1\,\,.\,3}}.\frac{9}{{2\,.\,4}}.\,\frac{{16}}{{3\,.\,5}}.\,\,...\,\,.\frac{{4\,\,092\,\,529}}{{2022\,.\,2024}}\]

\[ = \frac{{2\,.\,2}}{{1\,\,.\,3}}.\frac{{3\,.\,3}}{{2\,.\,4}}.\frac{{4\,.\,4}}{{3\,.\,5}}.\,\,...\,\,.\frac{{2023\,.\,2023}}{{2022\,.\,2024}}\]

\[ = \frac{{2\,.\,3\,.\,4.\,\,...\,\,.2023}}{{1\,.\,2\,.\,3.\,\,...\,\,.2022}}.\frac{{2\,.\,3\,.\,4.\,\,...\,\,.2023}}{{3\,.\,4\,.\,5.\,\,...\,\,.2024}}\]

\[ = \frac{{2023}}{1}.\frac{2}{{2024}} = \frac{{2023}}{{1012}}\].

Vậy \(A = \frac{{2023}}{{1012}}\).

a) \(\frac{5}{{17}} - \frac{{25}}{{31}} + \frac{{12}}{{17}} + \frac{{ - 6}}{{31}} = \left( {\frac{5}{{17}} + \frac{{12}}{{17}}} \right) + \left( {\frac{{ - 6}}{{31}} - \frac{{25}}{{31}}} \right)\)

\( = \left( {\frac{{5 + 12}}{{17}}} \right) + \left( {\frac{{ - 6 - 25}}{{31}}} \right) = \frac{{17}}{{17}} + \frac{{ - 31}}{{31}} = 1 + ( - 1) = 0\).

b) \(\frac{{17}}{8}:\left( {\frac{{23}}{8} + \frac{{ - 9}}{2}} \right) = \frac{{17}}{8}:\left( {\frac{{23}}{8} + \frac{{ - 36}}{8}} \right) = \frac{{17}}{8} & :\frac{{23 + \left( { - 36} \right)}}{8}\)

\( = \frac{{17}}{8} & :\frac{{ - 13}}{8} = \frac{{17}}{8} & .\frac{8}{{ - 13}} = \frac{{17}}{{ - 13}} = \frac{{ - 17}}{{13}}\).

c) \(\frac{3}{4}.\frac{{11}}{{27}} + \frac{{16}}{{27}}.\frac{3}{4} + \frac{1}{2}\)\[ = \frac{3}{4}.\left( {\frac{{11}}{{27}} + \frac{{16}}{{27}}} \right) + \frac{1}{2}\]

\[ = \frac{3}{4}.\frac{{27}}{{27}} + \frac{1}{2}\]\[ = \frac{3}{4} + \frac{1}{2} = \frac{3}{4} + \frac{2}{4} = \frac{5}{4}\].

d) \(\frac{{27}}{{23}} - \frac{{ - 5}}{{21}} - \frac{4}{{23}} + \frac{{16}}{{21}} + \frac{1}{2}\)\( = \left( {\frac{{27}}{{23}} - \frac{4}{{23}}} \right) + \left( {\frac{{16}}{{21}} - \frac{{ - 5}}{{21}}} \right) + \frac{1}{2}\)

\( = \left( {\frac{{27 - 4}}{{23}}} \right) + \left( {\frac{{16 + 5}}{{21}}} \right) + \frac{1}{2}\)\( = \frac{{23}}{{23}} + \frac{{21}}{{21}} + \frac{1}{2} = 1 + 1 + \frac{1}{2}\)

\( = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}\).