To find the area \( A \) of the square, we first simplify the expression for the side length \( s \):
The expression for the side length is given as:
\[ s = \frac{2x^2y}{xy^0} \]
Since \( y^0 = 1 \), we can simplify this to:
\[ s = \frac{2x^2y}{x} = 2xy \]
Next, we use the formula for the area \( A \) of a square, which is \( A = s^2 \):
\[ A = (2xy)^2 \]
Now we simplify \( (2xy)^2 \):
\[ A = 4x^2y^2 \]
Thus, the simplified form of the expression that represents the area is:
\[ \boxed{4x^2y^2} \]