To determine which measurement will require the use of irrational numbers, let's analyze each of the options given that the edge length \( a \) of the cube is a rational number:
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Volume of the cube: The volume \( V \) of a cube is given by the formula \( V = a^3 \). Since \( a \) is rational, \( a^3 \) will also be rational.
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Surface area of the cube: The surface area \( A \) of a cube is given by the formula \( A = 6a^2 \). Since \( a \) is rational, \( 6a^2 \) will also be rational.
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Area of one of the cube’s faces: The area of one face \( A_f \) of the cube is given by \( A_f = a^2 \). Since \( a \) is rational, \( a^2 \) will also be rational.
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Diagonal of one of the cube’s faces: The diagonal \( d \) of one face (which is a square) can be calculated using the Pythagorean theorem: \( d = a\sqrt{2} \). Here, \( \sqrt{2} \) is an irrational number, so \( d \) will be irrational because it is the product of the rational number \( a \) and the irrational number \( \sqrt{2} \).
Thus, the measurement that will require the use of irrational numbers is:
the diagonal of one of the cube’s faces.