To determine which measurement will require the use of irrational numbers, let’s analyze each option given that \( a \) is a rational number:
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The area of one of the cube’s faces:
- The area of one face of the cube is given by \( a^2 \), which is rational since the square of a rational number is also rational.
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The surface area of the cube:
- The surface area of the cube is given by \( 6a^2 \), which is also rational for the same reason as above.
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The volume of the cube:
- The volume of the cube is given by \( a^3 \), which is again rational since the cube of a rational number is rational.
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The diagonal of one of the cube’s faces:
- The diagonal \( d \) of one of the cube's faces 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, \( a\sqrt{2} \) will be irrational unless \( a = 0 \).
Therefore, the measurement that will require the use of irrational numbers is the diagonal of one of the cube’s faces.