### If a = 0

- If m is different from n, then either 0^(n-m) or 0^(m-n) is undefined (you cannot set zero to a negative power).
- If m equals n, then 0^0 is usually defined by convention to 1, and the expression becomes 1 / (1+1) + 1 / (1+1) = 1/2 + 1/2 = 1.

### If a is not zero

1 / (1+a^(n-m)) + 1 / (1+a^(m-n))

= (1+a^(m-n) + 1+a^(n-m)) / ( (1+a^(n-m))*(1+a^(m-n)) )

= (2 + a^(m-n) + a^(n-m)) / (1 + a^(n-m) + a^(m-n) +a^(n-m)*a^(m-n))

but: a^(n-m) = a^-(n-m) = 1 / a^(m-n)

thus: a^(n-m)*a^(m-n) = a^(n-m) / a^(m-n) = 1

We can conclude:

1 / (1+a^(n-m)) + 1 / (1+a^(m-n))

= (2 + a^(m-n) + a^(n-m)) / (2 + a^(n-m) + a^(m-n))

= 1

### Conclusion

If a=0 and m<>n, the expression is undefined, else it equals to 1.