To determine which measurement will require the use of irrational numbers, let’s analyze each option:
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The volume of the cube is calculated using the formula \( V = a^3 \). Since \( a \) is a rational number, \( a^3 \) will also be a rational number.
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The area of one of the cube’s faces is calculated using the formula \( A = a^2 \). Again, since \( a \) is a rational number, \( a^2 \) will also be a rational number.
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The surface area of the cube is calculated using the formula \( SA = 6a^2 \). Since \( a^2 \) is rational, \( 6a^2 \) will also be rational.
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The diagonal of one of the cube’s faces can be found using the formula \( d = a\sqrt{2} \). Here, \( \sqrt{2} \) is an irrational number. Therefore, \( a\sqrt{2} \) will be irrational, as a rational number multiplied by an irrational number results in an irrational number.
Based on this analysis, the measurement that will require the use of irrational numbers is:
the diagonal of one of the cube’s faces.
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