Exponents and Division

$\dfrac{12^{18}}{18^{12}}=\dfrac{(2^2\cdot 3)^{18}}{(2\cdot 3^2)^{12}}=\dfrac{2^{36}\cdot 3^{18}}{2^{12}\cdot 3^{24}}=\dfrac{2^{24}}{3^6}=\left(\dfrac{2^8}{3^2}\right)^3=\left(\dfrac{256}{9}\right)^3=\left(\dfrac{m}{n}\right)^3$

$\Longrightarrow m-n=256-9=\boxed{247}$

$\Large\frac{12^{18}}{18^{12}} = \frac{4^{18}\times3^{18}}{2^{12}\times 9^{12}}= \frac {2^{36}3^{18}}{2^{12}3^{24}} = \frac{2^{24}}{3^6} = (\frac{2^8}{3^2})^3= (\frac{256}{9})^3\implies 256 - 9=247$