The Closure Property states that a set is closed under an operation if performing that operation on members of the set always produces a member of the same set. For the set of rational numbers, this means that if you add, subtract, multiply, or divide (except by zero) two rational numbers, the result will also be a rational number.
Let's analyze each option:
A. \( \sqrt{36} + \sqrt{221} \)
- \( \sqrt{36} = 6 \) (a rational number)
- \( \sqrt{221} \) is not a perfect square, thus it is irrational.
- The sum \( 6 + \sqrt{221} \) is irrational.
- Not a rational number.
B. \( n + \sqrt{4} \)
- Assuming \( n \) is a rational number and \( \sqrt{4} = 2 \) (a rational number).
- The sum \( n + 2 \) is still rational.
- Is a rational number.
C. \( 0.6 + \frac{9}{10} \)
- \( 0.6 \) can be expressed as \( \frac{6}{10} \), which is a rational number.
- \( \frac{9}{10} \) is also a rational number.
- The sum \( 0.6 + \frac{9}{10} = \frac{6}{10} + \frac{9}{10} = \frac{15}{10} = 1.5 \), which is a rational number.
- Is a rational number.
D. \( 0.6479953469975... + \frac{3}{4} \)
- The number \( 0.6479953469975... \) appears to be a non-terminating decimal, and it's unclear whether it is rational or irrational without further context (it is likely irrational).
- \( \frac{3}{4} \) is a rational number.
- If \( 0.6479953469975... \) is irrational, the sum would also be irrational.
- Cannot conclude this is a rational number without knowing the nature of the decimal.
Conclusion:
The options that result in a rational number are:
- B. (If \( n \) is rational)
- C.
Thus, the sums that result in a rational number are B and C.
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