To identify the measurement that will require the use of irrational numbers, let's analyze each option:
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Surface area of the cube: The surface area \( S \) of a cube with edge length \( a \) is given by \( S = 6a^2 \). Since \( a \) is a rational number, \( a^2 \) will also be rational, and thus the surface area will be rational.
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Area of one of the cubeā€™s faces: The area \( A \) of one face of the cube is \( A = a^2 \). Again, 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 \( V = a^3 \). Since \( a \) is rational, \( a^3 \) will also be rational.
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Diagonal of one of the cubeā€™s faces: The diagonal \( d \) of one face of the cube can be calculated using the Pythagorean theorem, which gives \( d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \). Here, \( a\sqrt{2} \) involves \( \sqrt{2} \), which is an irrational number. Thus, the diagonal will be an irrational number.
Based on this analysis, the measurement that will require the use of irrational numbers is:
the diagonal of one of the cubeā€™s faces.