Introduction:
In order for a physical quantity to be dimensionless, it must have no units associated with it. In this case, we have three physical quantities: radiation pressure (P), speed of light (c), and radiation energy striking per unit area per second (S). We need to find the values of the non-zero integers x, y, and z such that P^x, S^y, and c^z are dimensionless.

Explanation:

Dimension of Radiation Pressure (P):
The dimension of pressure is force per unit area. Thus, the dimension of radiation pressure can be expressed as:
[P] = [F/A]

Dimension of Speed of Light (c):
The dimension of speed is distance per unit time. Thus, the dimension of speed of light can be expressed as:
[c] = [L/T]

Dimension of Radiation Energy Striking per Unit Area per Second (S):
The dimension of energy is force times distance. Thus, the dimension of energy striking per unit area per second can be expressed as:
[S] = [F*L^2/T*A]

Requirement for Dimensionless Quantity:
For a quantity to be dimensionless, the dimensions of the numerator and denominator must cancel out. In other words, the powers of all base dimensions should add up to zero.

Analysis:
Let's analyze the dimensions of each quantity separately and determine the values of x, y, and z that make them dimensionless.

Dimension of P^x:
[P^x] = [P]^x = ([F/A])^x
For [P^x] to be dimensionless, the dimensions of force (F) and area (A) should cancel out.
Thus, x should be 0.

Dimension of S^y:
[S^y] = [S]^y = ([F*L^2/T*A])^y
For [S^y] to be dimensionless, the dimensions of force (F), length (L), time (T), and area (A) should cancel out.
Thus, y should be 0.

Dimension of c^z:
[c^z] = [c]^z = ([L/T])^z
For [c^z] to be dimensionless, the dimensions of length (L) and time (T) should cancel out.
Thus, z should be 0.

Conclusion:
To make P^x, S^y, and c^z dimensionless, the non-zero integers x, y, and z should all be 0. This means that P^0, S^0, and c^0 will have no units associated with them and will be dimensionless.