To determine which measurement will require the use of irrational numbers, let's analyze the options in the context of a cube with edge length \( a \):
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The volume of the cube: The volume \( V \) of a cube is calculated using the formula \( V = a^3 \). Since \( a \) is a rational number, \( a^3 \) will also be rational.
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The 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, \( d \) will be irrational when \( a \) is a non-zero rational number.
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The surface area of the cube: The surface area \( SA \) of a cube is given by \( SA = 6a^2 \). Both \( 6 \) and \( a^2 \) are rational, so the surface area is rational.
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The area of one of the cube’s faces: The area \( A \) of one face of the cube is \( A = a^2 \). Since \( a \) is rational, \( a^2 \) is also rational.
Based on this analysis, the measurement that requires the use of irrational numbers is:
the diagonal of one of the cube’s faces.