The order and degree of $\left(\dfrac{d^2y}{dx^2}\right)^{1/2} - \left(\dfrac{dy}{dx}\right)^{1/3} = 20$ are:

Raise to suitable power to clear radicals. Let $u = (d^2y/dx^2)^{1/2}$ and $v = (dy/dx)^{1/3}$: $u - v = 20$ ⇒ $u = 20 + v$, square: $(d^2y/dx^2) = (20+v)^2$. To clear the cube root in $v = (dy/dx)^{1/3}$, raise both sides to 6th power (LCM of 2 and 3). Result: order 2 (highest derivative is $d^2y/dx^2$), degree 6 (after polynomialising).