Official solution
Tr+1 = 18Cr \[{\left( {{x^{{1 \over 3}}}} \right)^{18 - r}}\] . \[{\left( {{1 \over {2{x^{{1 \over 3}}}}}} \right)^r}\]
= 18Cr \[{\left( {{1 \over 2}} \right)^r}\,\,.\,\,{x^{{{18 - 2r} \over 3}}}\]
For coefficient of x-2,
\[{{18 - 2r} \over 3}\] = -2
\[ \Rightarrow \] r = 12
\[ \therefore \] Coefficient of x-2 is (m) = 18C12 \[{\left( {{1 \over 2}} \right)^{12}}\]
For coefficient of x-4,
\[{{18 - 2r} \over 3}\] = - 4
\[ \Rightarrow \] r = 15
\[ \therefore \] Coefficient of x-4 is (n) = 18C15 \[\left( {{1 \over {2}}} \right)\]15
\[ \therefore \] \[{m \over n} = {{^{18}{C_{12}}{{\left( {{1 \over 2}} \right)}^{12}}} \over {^{18}{C_{15}}{{\left( {{1 \over 2}} \right)}^{15}}}}\]
= \[{{{}^{18}{C_6} \times {{\left( 2 \right)}^3}} \over {{}^{18}{C_3}}}\]
= 182