m,n  are the smallest positive integers satisfying the relation ${\left(2cis\frac{\pi }{6}\right)}^m={\left(4cis\frac{\pi }{4}\right)}^n$ where $i=\sqrt{-1}$  then $m+n$ equals to

$cis\frac{\pi }{6}=\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}=\frac{\sqrt{3}}{2}+i\frac{1}{2}=\frac{\sqrt{3}+i}{2}\times \frac{i}{i}$

$=\frac{-1+i\sqrt{3}}{2}\times \frac{1}{i}=\frac{w}{i}=-iw$

${\left(2cis\frac{\pi }{6}\right)}^m={(-2iw)}^m={\left[{(-2iw)}^3\right]}^{\frac{m}{3}}={\left(8i\right)}^{\frac{m}{3}}$

${\left(4cis\frac{\pi }{4}\right)}^n={\left[4(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4})\right]}^n={\left[2\sqrt{2}(1+i)\right]}^n$

$={\left\{{\left[8(1+i)\right]}^2\right\}}^{\frac{n}{2}}={\left(16i\right)}^{\frac{n}{2}}$

$\therefore {\left(8i\right)}^{\frac{m}{3}}={\left(16i\right)}^{\raisebox{1ex}{$n$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

Which is satisfy only when $m=48$ and  $n=24$ $m+n=72$