To solve the equation 7^2*X - 6 = 49, we can follow these steps:

1. Simplify the left side of the equation by calculating 7^2, which is 49. The equation becomes 49*X - 6 = 49.

2. Add 6 to both sides of the equation to isolate the term with X. The equation becomes 49*X = 55.

3. Divide both sides of the equation by 49 to solve for X. The equation becomes X = 55/49.

4. Simplify the fraction 55/49 by dividing both the numerator and denominator by their greatest common divisor, which is 11. The equation becomes X = 5/7.

Therefore, the solution to the equation 7^2*X - 6 = 49 is X = 5/7.