To identify which measurement will require the use of irrational numbers for a cube with an edge length \( a \) that is a rational number, we will consider the following expressions:
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Surface Area of the Cube: The surface area \( S \) of a cube is given by the formula \( S = 6a^2 \). Since \( a \) is rational, \( a^2 \) is also rational, and thus the surface area is rational.
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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, \( a^3 \) is also rational, so the volume is rational.
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Area of One of the Cube’s Faces: The area \( A \) of one face of the cube is given by \( A = a^2 \). As with the surface area, since \( a \) is rational, \( a^2 \) is rational, so the area of one face is rational.
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Diagonal of One of the Cube’s Faces: The diagonal \( d \) of one face of the cube (which is a square) can be computed using the Pythagorean theorem: \( d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \). Since \( a \) is rational, \( d \) will be irrational because \( \sqrt{2} \) is an irrational number.
Therefore, the measurement that will require the use of irrational numbers is:
The diagonal of one of the cube’s faces.