To determine which measurement will require the use of irrational numbers, let's analyze each option:
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The surface area of the cube: The formula for the surface area (SA) of a cube is \( SA = 6a^2 \), where \( a \) is a rational number. Since squaring a rational number results in a rational number, the surface area will also be rational.
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The volume of the cube: The formula for the volume (V) of a cube is \( V = a^3 \). As with surface area, cubing a rational number yields a rational number, so the volume is also rational.
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The diagonal of one of the cube’s faces: The formula for the diagonal (d) of a square face of the cube is \( d = a\sqrt{2} \). Since \( \sqrt{2} \) is an irrational number, the diagonal will be irrational as long as \( a \) is a non-zero rational number.
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The area of one of the cube’s faces: The area (A) of one face of the cube is given by \( A = a^2 \), which is rational since squaring a rational number yields a rational number.
Among these options, the measurement that requires the use of irrational numbers is:
the diagonal of one of the cube’s faces.