Generalised Taxicab Numbers
05AB1E, 15 bytes
∞.ΔL¹m²ã€{ÙOy¢Q

Try it online!
This is really slow, inserting ¹zmï after .Δ limits the search space significantly: Try it online!
Commented:

                # Inputs ¹=i, ²=x, ³=n
∞.Δ             # find the first natural number k that satisfies:
   L            #   each of the range [1..k]
    ¹m          #   raised to the i-th power
      ²ã        #   take the list to the x-th cartesian power
        €{      #   sort each x-tuple
          Ù     #   deduplicate the tuples
           O    #   sum each x-tuple
            y¢  #   count the number of k's
              Q #   is this equal to the last input?

ã€{Ù does the same as Python's itertools.combinations_with_replacement, but there doesn't seem to be a builtin for this in 05AB1E.