If f(z)=m^z - n^z, and f(1)=2, f(2)=8. find f(3).

If f(z)=m^z - n^z and f(1)=2, f(2)=8
then
m -  n = 2 so m = n + 2 
m^2 - n^2 = 8

substitute m = n + 2  into m^2 - n^2 = 8

so
m^2 - n^2 = 8
(n + 2)^2 - n^2 = 8
n^2 + 4n + 4 - n^2 = 8
4n = 4
  n = 1

m = n + 2 = 1 + 2 = 3

so f(z) = 3^z - 1^z

if z = 1 the f(1) = 3^1 - 1^1 = 2
if z = 2 the f(2) = 3^2 - 1^2 = 9 - 1 = 8
if z = 3 the f(3) = 3^3 - 1^2 = 27 - 1 = 26

answer
f(3) = 26