To determine which measurement will require the use of irrational numbers in the context of a cube with edge length \( a \) (where \( a \) is a rational number), we will analyze each option:
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The area of one of the cube’s faces: The area of one face of a cube is calculated as \( a \times a = a^2 \). Since \( a^2 \) is a rational number (the square of a rational number is rational), this measurement does not involve irrational numbers.
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The volume of the cube: The volume of the cube is calculated as \( a \times a \times a = a^3 \). Since \( a^3 \) is also a rational number (the cube of a rational number is rational), this measurement does not involve irrational numbers either.
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The diagonal of one of the cube’s faces: The diagonal of a square face can be calculated using the Pythagorean theorem: \( d = \sqrt{a^2 + a^2} = \sqrt{2a^2} = a\sqrt{2} \). Here, \( \sqrt{2} \) is an irrational number. Therefore, \( d \) is a product of a rational number \( a \) and an irrational number \( \sqrt{2} \), making \( d \) irrational.
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The surface area of the cube: The surface area of the cube is calculated as \( 6 \times a^2 \). Since \( a^2 \) is rational, \( 6a^2 \) is also a rational number. Hence, this measurement does not involve 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.