To identify which measurement will require the use of irrational numbers when the cube has an edge length \( a \) that is a rational number, we can analyze each option:
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The area of one of the cube’s faces: The area \( A \) of a face of the cube is given by \( A = a^2 \). Since \( a \) is rational, \( a^2 \) is also rational.
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The surface area of the cube: The surface area \( S \) of the cube is \( S = 6a^2 \). Again, since \( a^2 \) is rational, \( S \) is also rational.
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The volume of the cube: The volume \( V \) of the cube is \( V = a^3 \). Since \( a \) is rational, \( a^3 \) is also rational.
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The diagonal of one of the cube’s faces: The diagonal \( d \) of one of the cube's faces is calculated using the Pythagorean theorem: \[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}. \] Since \( a \) is rational, \( d \) involves \( \sqrt{2} \), which is an irrational number.
Thus, the measurement that will require the use of irrational numbers is the diagonal of one of the cube’s faces.