Differential Equation:
The given differential equation is (d²y/dx²)^(2/3) = (y * (dy/dx))^(1/2).

Order of the Differential Equation:
The order of a differential equation is determined by the highest derivative present in the equation. In this case, the highest derivative is d²y/dx², which means the equation is at least a second-order differential equation.

Degree of the Differential Equation:
The degree of a differential equation is determined by the highest power of the highest derivative present in the equation. In this case, the highest power of d²y/dx² is 2/3, which means the equation is at least a fourth-degree differential equation.

Analysis:
To determine the exact order and degree of the given differential equation, let's rewrite it in a more standard form.

Taking the square of both sides, we get:
(d²y/dx²)^(4/3) = (y * (dy/dx))^(1/2)^2
(d²y/dx²)^(4/3) = y * (dy/dx)

Now, let's simplify and differentiate both sides of the equation with respect to x.

Differentiating the left side:
d/dx [(d²y/dx²)^(4/3)] = d/dx [y * (dy/dx)]
(4/3) * (d²y/dx²)^(1/3) * (d²y/dx²) = (dy/dx) * (dy/dx) + y * (d²y/dx²)

Simplifying the above equation:
(4/3) * (d²y/dx²)^(1/3) * (d²y/dx²) = (dy/dx)² + y * (d²y/dx²)

Now, let's analyze the terms present in the equation:

- (d²y/dx²) is a second-order derivative.
- (dy/dx)² is a first-order derivative squared.
- y * (d²y/dx²) is a product of a function and a second-order derivative.

Conclusion:
From the analysis above, we can conclude that the given differential equation is a second-order differential equation because it contains a second-order derivative term. Additionally, the highest power of the highest derivative is 2/3, making it a fourth-degree differential equation.

Hence, the correct answer is option 'A' - second order, fourth degree.