\(Rat{e_2} = \,k{[2A]^n}{\left[ {\frac{1}{2}B} \right]^m}\)
\(\therefore \,\frac{{Rat{e_2}}}{{Rat{e_1}}}\, = \,\frac{{k{{[2A]}^n}{{\left[ {\frac{1}{2}B} \right]}^m}}}{{k\,{{[A]}^n}{{[B]}^m}}}\, = \,{(2)^n}{\left( {\frac{1}{2}} \right)^m}\)
\( = \,{(2)^n}.{(2)^{ - m}}\, = \,{2^{n - m}}\)