Inspiration #1

The number $6$ has $4$ factors, namely $1$ , $2$ , $3$ and $6$ .

As such , 4 cases are classified:

Case i: $m$ is already a power of $6$ . There are $4$ such numbers (1, 64, 729, 4096). This expression is also equivalent to $(x^{6}) ^ {n}$ .

The number of expressions is 4 x 7102 = 28408.

Case ii: $m$ is a power of $3$ (but not a power of $2$ ). There are $\lfloor \sqrt[3] 7102 \rfloor$ = $19$ such numbers. Excluding the past 4 powers of $6$ , we have $15$ such numbers. This expression is also equivalent to $(x^[3])^[n])$ . In order to have a sixth power, $n$ must be a multiple of $2$ . There are $\lfloor \frac{ 7102 }{ 2 } \rfloor$ = $3551$ value for $n$ .

The number of expressions is 15 x 3551 = 53265.

Case iii: $m$ is a power of $2$ (but not a power of $3$ ). There are $\lfloor \sqrt[2] 7102 \rfloor$ = $84$ such numbers. Excluding the past 4 powers of $6$ , we have $80$ such numbers. This expression is also equivalent to $(x^[2])^[n])$ . In order to have a sixth power, $n$ must be a multiple of $3$ . There are $\lfloor \frac{ 7102 }{ 3 } \rfloor$ = $2367$ value for $n$ .

The number of expressions is 80 x 2367 = 189360.

Case iv: $m$ is a power of $1$ (but not a power of $2 , 3 , 6$ ). There are $7102$ such numbers. Excluding the past (80 + 15 + 4) powers of $2$ , $3$ and $6$ , we have $7003$ such numbers. This expression is also equivalent to $(x^[1])^[n])$ . In order to have a sixth power, $n$ must be a multiple of $6$ . There are $\lfloor \frac{ 7102 }{ 6 } \rfloor$ = $1183$ value for $n$ .

The number of expressions is 7003 x 1183 = 8284549.

There are in total $28408 + 53265 + 189360 + 8284549$ = $8555582$ expressions. Its digit sum is $8+5+5+5+5+8+2$ is $38$