More mean numbers

How many seven-digit positive numbers, containing no zeroes, are there such that the geometric mean of the first six digits is equal to the seventh digit?

In other words, how many distinct numbers $\overline{d_1d_2\ldots d_6d_7}$ are there such that $d_1\times d_2\times \cdots \times d_6 = (d_7)^6,$ where $1 \leq d_i \leq 9?$

It is probably easier to write code to solve this problem. However, a mathematical solution using combinatorics might be more interesting!

This is a follow-up on this problem, but more elaborate.