Generalised Taxicab Numbers \$\newcommand{T}[1]{\text{Ta}(#1)} \newcommand{Ta}[3]{\text{Ta}_{#2}^{#3}(#1)} \T n\$ is a function which returns the smallest positive integer which can be expressed as the sum of 2 positive integer cubes in \$n\$ different ways. For example, \$\T 1 = 2 = 1^3 + 1^3\$ and \$\T 2 = 1729 = 1^3 + 12^3 = 9^3 + 10^3\$ (the Hardy-Ramanujan number).
Let's generalise this by defining a related function: \$\Ta n x i\$ which returns the smallest positive integer which can be expressed as the sum of \$x\$ \$i\$th powers of positive integers in \$n\$ different ways. In this case, \$\T n = \Ta n 2 3\$ (note: this is the same function here, \$\Ta n x i = \text{Taxicab}(i, x, n)\$)
Your task is to take 3 positive integers \$n, x\$ and \$i\$ and return \$\Ta n x i\$. This is code-golf so the shortest code in bytes wins.
In case \$x = 1 \$ and \$ n &gt; 1\$, your program can do anything short of summoning Cthulhu, and for other cases where \$\Ta n x i\$ is not known to exist (e.g. \$\Ta n 2 5\$), the same applies.
Test cases
n, x, i -&gt; out
1, 1, 2 -&gt; 1
1, 2, 3 -&gt; 2
2, 2, 2 -&gt; 50
2, 2, 3 -&gt; 1729
3, 3, 2 -&gt; 54
3, 3, 3 -&gt; 5104
2, 6, 6 -&gt; 570947
2, 4, 4 -&gt; 259
6, 4, 4 -&gt; 3847554
2, 5, 2 -&gt; 20
2, 7, 3 -&gt; 131
2, 5, 7 -&gt; 1229250016
5, 8, 4 -&gt; 4228


Properties of \$\Ta n x i\$

\$\forall i : \Ta 1 x i = x\$ as \$x = \underbrace{1^i + \cdots + 1^i}_{x \text{ times}}\$
\$\Ta n 1 i\$ does not exist for all \$n &gt; 1\$
\$\Ta n 2 5\$ is not known to exist for any \$n \ge 2\$

This is a table of results \$\{\Ta n x i \:|\: 1 \le n,x,i \le 3 \}\$, ignoring \$\Ta 2 1 i\$ and \$\Ta 3 1 i\$:
$$\begin{array}{ccc|c}
n &amp; x &amp; i &amp; \Ta n x i \\
\hline
1 &amp; 1 &amp; 1 &amp;  1 \\
1 &amp; 1 &amp; 2 &amp;  1 \\
1 &amp; 1 &amp; 3 &amp;  1 \\
1 &amp; 2 &amp; 1 &amp;  2 \\
1 &amp; 2 &amp; 2 &amp;  2 \\
1 &amp; 2 &amp; 3 &amp;  2 \\
1 &amp; 3 &amp; 1 &amp;  3 \\
1 &amp; 3 &amp; 2 &amp;  3 \\
1 &amp; 3 &amp; 3 &amp;  3 \\
2 &amp; 2 &amp; 1 &amp;  4 \\
2 &amp; 2 &amp; 2 &amp;  50 \\
2 &amp; 2 &amp; 3 &amp;  1729 \\
2 &amp; 3 &amp; 1 &amp;  5 \\
2 &amp; 3 &amp; 2 &amp;  27 \\
2 &amp; 3 &amp; 3 &amp;  251 \\
3 &amp; 2 &amp; 1 &amp;  6 \\
3 &amp; 2 &amp; 2 &amp;  325 \\
3 &amp; 2 &amp; 3 &amp;  87539319 \\
3 &amp; 3 &amp; 1 &amp;  6 \\
3 &amp; 3 &amp; 2 &amp;  54 \\
3 &amp; 3 &amp; 3 &amp;  5104 \\
\end{array}$$
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.