We have

$\begin{array}{r}(w+x{)}^n{=}^n{C}_0{w}^n{x}^0{+}^n{C}_1{w}^{n-1}x{+}^n{C}_2{w}^{n-2}{x}^2\\ +\dots {+}^n{C}_n{x}^n\end{array}$

$\therefore w^n=1,n{w}^{n-1}x=12\overline{w}$and ${}^n{C}_2{w}^{n-2}{x}^2=69w$

Now, 

$w^n=1\Rightarrow n=3m,m\in N$

$\therefore n{w}^{n-1}x=12\overline{w}$and ${ }^n{C}_2{w}^{n-2}{x}^2=69w$

$\Rightarrow 3mx=12$and $\frac{3m(3m-1)}{2}{x}^2=69$

$\Rightarrow mx=4$and $3m^2{x}^2-\left(mx\right)x=46$

$\Rightarrow mx=4$and $3\times 16-4x=46$

$\Rightarrow x=\frac{1}{2}$and $m=8$

$\Rightarrow x=\frac{1}{2}$and $n=24$                     $[\because n=3m]$