Explanation:
It is in fact a very interesting question.
the "square-root" is a function which returns a single positive value (call the principal square-root to help distinguish from the popular notion that square-root can be positive or negative (see later paragraph).
Thus the square-root (or any even powered root) of any positive number is the principal (positive) root of the positive number.
In principle, we do not need the absolute value sign because the principal root is always positive. However, to avoid confusion, it is not wrong to add an absolute value sign/function to clarify our intention/interpretation.
From the above discussion, we see that the value of a cube-root (as in case I) takes the sign of the argument. In this case, it depends on the sign of x, so we cannot use the absolute value sign to qualify the result.
In the other cases (II, III and IV), the powers of x and y are all even, so assuming they are real numbers, their product would be positive. So the principal square-root (and the even root of 4 (which is square-root twice) are positive, so applying the absolute-value sign/function will not hurt, although not mathematically necessary, as they are just an expression, not part of an equation.
As to from where did the "popular notion" that the square-root can be positive or negative? It roots from a confusion between the square-root of x^2 and the solution to the equation x=3^2. The former is a positive number, while the latter is ±3. So when it is the square-root sign itself within an expression, not within an equation, the value is a positive number.