To analyze which measurement will require the use of irrational numbers, we can examine each option:
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The area of one of the cube’s faces: The area \(A\) of a face of a cube with edge length \(a\) is given by the formula \(A = a^2\). Since \(a\) is a rational number, \(a^2\) will also be a rational number. Therefore, the area of one of the cube's faces does not require irrational numbers.
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The diagonal of one of the cube’s faces: The diagonal \(d\) of a face of a cube can be calculated using the Pythagorean theorem. If the face is a square with edge length \(a\), then the diagonal is \(d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2}\). Since \(\sqrt{2}\) is an irrational number, the diagonal of one of the cube’s faces will require the use of an irrational number if \(a\) is rational.
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The surface area of the cube: The surface area \(SA\) of a cube is given by \(SA = 6a^2\). Since \(a^2\) is rational, \(6a^2\) will also be rational. Therefore, the surface area of the cube does not require the use of irrational numbers.
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The volume of the cube: The volume \(V\) of a cube is given by \(V = a^3\). Since the cube of a rational number is also rational, the volume of the cube does not require the use of irrational numbers.
Based on this analysis, the measurement that will require the use of irrational numbers is:
the diagonal of one of the cube’s faces.