To determine which measurement will require the use of irrational numbers, let's analyze each option:
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Surface area of the cube: The formula is \(6a^2\), which is a rational number since both 6 and \(a^2\) are rational (given that \(a\) is a rational number).
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Volume of the cube: The formula is \(a^3\), which is also rational since a rational number raised to any power remains rational.
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Diagonal of one of the cube’s faces: The formula for the diagonal \(d\) of a cube's face (which is a square) is given by \(d = a\sqrt{2}\). Since \(\sqrt{2}\) is an irrational number, the diagonal will be irrational when multiplied by \(a\).
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Area of one of the cube’s faces: The area is calculated as \(a^2\), which is rational.
Considering these points, the measurement that will require the use of irrational numbers is:
the diagonal of one of the cube’s faces.