To determine which measurement will require the use of irrational numbers for a cube with a rational edge length \( a \):
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Area of one of the cube’s faces: The area of a face is calculated as \( a^2 \), which remains a rational number.
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Diagonal of one of the cube’s faces: The diagonal \( d \) of a face can be found using the formula \( d = a\sqrt{2} \). Since \( \sqrt{2} \) is an irrational number, the diagonal of the face will be irrational if \( a \) is rational.
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Surface area of the cube: The surface area is given by \( 6a^2 \), which is rational since it is based on the square of \( a \).
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Volume of the cube: The volume is calculated as \( a^3 \), which is also rational.
Thus, the measurement that will require the use of irrational numbers is the diagonal of one of the cube’s faces.