Here's a cross-number puzzle I came across recently.

Here's a cross-number puzzle I came across recently.

$\displaystyle \hspace{50pt}\text{Clues} \\ \underline{\text{Across}} \hspace{70pt} \underline{\text{Down}} \\ 1\quad a^{2}\hspace{75pt}1\quad e \\ 5 \quad b^{3} \hspace{75pt}2 \quad f^{2} \\ 6 \quad c^{4} \hspace {75pt} 3 \quad g \\ 7 \quad d^{4} \hspace{75pt} 4 \quad h^{4}$

The letters a, b, c, d, e, f, g and h denote distinct positive integers. 

$\begin{array}{|l|l|l|l|} \hline 1 \hspace{20pt} & 2 \hspace{20pt} & 3 \hspace{20pt} & 4 \hspace{20pt} \\ & & & \\ \hline 5 & & & \\ & & & \\ \hline 6 & & & \\ & & & \\ \hline 7 & & & \\ & & & \\ \hline \end{array}$

The answer to each clue is a 4-digit number, the first digit of which is not zero.

Show that your answer to the complete puzzle is not unique, but that the sum e + g is unique, and calculate its value.