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Choice A is correct. It's given that the area of a square is $64$ square inches. The area $A$, in square inches, of a square is given by the formula $A=s^2$, where $s$ is the side length, in inches, of the square. Substituting $64$ for $A$ in this formula yields $64=s^2$. Taking the positive square root of both sides of this equation yields $8=s$. Thus, the side length, in inches, of this square is $8$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This is the area, in square inches, of the square, not the side length, in inches, of the square.

Choice D is incorrect and may result from conceptual or calculation errors.