Let $A=\{1,2,3,\dots .7\}$ and let $P\left(A\right)$denote the power set of $A$. If the number of functions $f:A\to P\left(A\right)$ such that $a\in f\left(a\right),\forall a\in A$ is $m^n,m$ and $n\in N$ and $m$ is least, then $m+n$ is equal to ______.

Given,

$A=\{1,2,3,\dots .7\}$

And $f:A\to P\left(A\right)$ and $a\in A$

Also $a\in f\left(a\right)$, that means $‘a’$ will be the element of subset as $f:A\to P\left(A\right)$,

Now, total options for $1$ will be $2^6$. (Because $2^6$ subsets contains elements$1$)

Similarly, for every other element we get, total  $2^6\times 2^6\times 2^6\times 2^6\times 2^6\times 2^6\times 2^6=2^{42}$ subsets.

Now, comparing with $m^n$ we get, $m=2 \& n=42$

Hence,  $m+n=2+42=44$