What???

Let rewrite the equation to:

${ m }^{ n }+{ m }^{ n+1 }+{ m }^{ n+2 }={ m }^{ n }(1+m+{ m }^{ 2 })$

So ${ m }^{ n }$ is a factor of 39, so ${ m }^{ n }=1,3,13,39$ . Analyzing all the possibilities and considering that $\frac { 39 }{ { m }^{ n } } =1+m+{ m }^{ 2 }$ , so $m=3$ and $n=1$ .

Therefore ${ n }^{ m }={ 1 }^{ 3 }=1$