To solve the equation \((X - 7)^7 = 9\), we can follow these steps:
- Take the seventh root of both sides to eliminate the exponent:
\[ X - 7 = 9^{1/7} \]
- Now, solve for \(X\) by adding \(7\) to both sides:
\[ X = 9^{1/7} + 7 \]
The seventh root of \(9\) is the value we obtained. Thus, the solution to the equation is:
\[ X = 7 + 9^{1/7} \]
This is the primary solution. If you're looking for real solutions, this is the only one because raising to an odd power maintains the sign. If complex solutions are considered, they would involve obtaining all seventh roots of \(9\), but for real solutions, we stick with the one found.
So, the final answer is:
\[ X = 7 + 9^{1/7} \]