To determine which measurement will require the use of irrational numbers for a cube with a rational edge length \( a \), let's analyze each option:
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Surface area of the cube: The formula for the surface area of a cube is \( 6a^2 \). Since both \( 6 \) and \( a^2 \) (where \( a \) is rational) are rational, the surface area will also be a rational number.
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Diagonal of one of the cube’s faces: The diagonal \( d \) of a face of the cube can be calculated using the Pythagorean theorem: \[ d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}. \] Since \( \sqrt{2} \) is an irrational number, the diagonal of a face of the cube will be an irrational number if \( a \) is rational.
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Area of one of the cube’s faces: The area \( A \) of a face is given by \( a^2 \). Since \( a \) is rational, \( a^2 \) will also be rational.
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Volume of the cube: The volume \( V \) of the cube is given by \( a^3 \). Since \( a \) is rational, \( a^3 \) will also be a rational number.
Based on this analysis, the measurement that will require the use of irrational numbers is:
the diagonal of one of the cube’s faces.