To determine the nature of the product of \( \frac{1}{4} \cdot 10.2568 \), we first analyze the numbers involved.
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\( \frac{1}{4} \) is a rational number (it can be expressed as the fraction of two integers).
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\( 10.2568 \) is a decimal number, which can also be expressed as a fraction (it's \( \frac{102568}{10000} \)), making it a rational number as well.
The product of two rational numbers is always a rational number. Therefore, \( \frac{1}{4} \cdot 10.2568 \) is also a rational number.
The correct response is:
Rational, because both numbers are rational.