Cho các số thực \(x\), \(y\) thỏa \({5^x} = \frac{1}{2},\,\,\,{5^y} = \frac{1}{3}\). Khi đó giá trị biểu thức \(\frac{{{2^x}{{.10}^y} + {2^y}{{.10}^x}}}{{{{10}^{x + y}}}}\) bằng

\(A = \frac{{{{\left( {{{10}^2}} \right)}^\alpha } \cdot {{\left( {{{10}^{ - 3}}} \right)}^\beta }}}{{{{10}^{ - \alpha }} \cdot {{10}^{2\beta }}}} = \frac{{{{10}^{2\alpha }}}}{{{{10}^{ - \alpha }}}} \cdot \frac{{{{10}^{ - 3\beta }}}}{{{{10}^{2\beta }}}} = {10^{2\alpha  - ( - \alpha )}} \cdot {10^{ - 3\beta  - 2\beta }}\)

\( = {10^{3\alpha }} \cdot {10^{ - 5\beta }} = {\left( {{{10}^\alpha }} \right)^3} \cdot {\left( {{{10}^\beta }} \right)^{ - 5}} = {3^3} \cdot {7^{ - 5}} = \frac{{{3^3}}}{{{7^5}}} = \frac{{27}}{{16807}}{\rm{. }}\)

\({\left( {\frac{3}{4}} \right)^{ - 2}}{.3^2}{.12^0} = \frac{{{4^2}}}{{{3^2}}}{.3^2}.1 = 16\).