Final answer:
Multiplying by a fraction and multiplying a fraction by a repeating decimal both result in a rational number. Adding the square root of a perfect square to a whole number also yields a rational result.
Step-by-step explanation:
The question posed pertains to operations that result in a rational answer. To understand which operation yields a rational result, let's explore each option:
- Multiplying by a fraction: When you multiply any number by a fraction, assuming the fraction is rational (both numerator and denominator are integers), the result is also rational. This operation does not change the intrinsic nature of the number's rationality.
- Adding the square root of a nonperfect square to a whole number: This operation typically results in an irrational number as the square root of a nonperfect square is irrational, and adding an irrational number to a rational number (a whole number) gives an irrational result.
- Adding the square root of a perfect square to a whole number: Since the square root of a perfect square is a whole number, adding it to another whole number gives a rational number.
- Multiplying a fraction by a repeating decimal: A repeating decimal is a rational number. Therefore, multiplying it by a rational fraction will also yield a rational result.
In summary, options: multiplying by a fraction and multiplying a fraction by a repeating decimal will definitely result in a rational number. Additionally, adding the square root of a perfect square to a whole number will also result in a rational number, but adding the square root of a nonperfect square will not.