To determine the nature of the product of \( \frac{1}{4} \) and \( 10.2568 \), let's analyze the two numbers involved:
- \( \frac{1}{4} \) is a rational number.
- \( 10.2568 \) is also a rational number. Any decimal number that terminates (like 10.2568) can be expressed as a fraction, making it rational.
Now let's perform the multiplication:
\[ \frac{1}{4} \times 10.2568 = \frac{10.2568}{4} \]
To check if this results in an integer, we can divide \( 10.2568 \) by \( 4 \):
\[ 10.2568 \div 4 = 2.5642 \]
Since \( 2.5642 \) is not an integer, the product of \( \frac{1}{4} \times 10.2568 \) is not an integer.
Summary of options:
- Integer: No, it's not an integer.
- Because both numbers are rational: While it's true that both numbers are rational, this does not define the product.
- Radical: This is irrelevant as there are no radicals involved.
- Irrational because one number is irrational: This statement is false; both numbers are rational.
Thus, none of the provided statements accurately describe the product. However, the product itself is rational, as it is the product of two rational numbers.