To solve the equation 5(x-4)^2 -7=13, we will follow these steps:

Step 1: Add 7 to both sides of the equation to isolate the squared term.
5(x-4)^2 = 20

Step 2: Divide both sides of the equation by 5 to isolate the squared term.
(x-4)^2 = 4

Step 3: Take the square root of both sides of the equation.
√((x-4)^2) = √4

Step 4: Simplify the square root on the left side of the equation.
|x-4| = 2

Step 5: Set up two equations, one with a positive value and one with a negative value.
x-4 = 2 or x-4 = -2

Step 6: Solve each equation separately.
For x-4 = 2:
x = 2 + 4
x = 6

For x-4 = -2:
x = -2 + 4
x = 2

Therefore, the solutions to the equation 5(x-4)^2 -7=13 are x = 6 and x = 2.