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. In this case, we are considering sums that yield rational numbers.
Let's evaluate each option:
A. \(\sqrt{36} + \sqrt{221}\)
- \(\sqrt{36} = 6\) (which is rational)
- \(\sqrt{221}\) is irrational.
- Therefore, \(6 + \sqrt{221}\) results in an irrational number.
B. \(\pi + \sqrt{4}\)
- \(\sqrt{4} = 2\) (which is rational)
- \(\pi\) is irrational.
- Therefore, \(\pi + 2\) results in an irrational number.
C. \(0.6 + \frac{9}{10}\)
- \(0.6\) is equivalent to \(\frac{6}{10}\) or \(\frac{3}{5}\) (which is rational).
- \(\frac{9}{10}\) is also rational.
- Therefore, \(0.6 + \frac{9}{10} = \frac{6}{10} + \frac{9}{10} = \frac{15}{10} = 1.5\), which is rational.
D. \(0.6479953469975... + \frac{3}{4}\)
- \(0.6479953469975...\) is a decimal number that does not terminate or repeat exactly. It is not a rational number (infinite non-repeating decimal).
- \(\frac{3}{4}\) is rational.
- Therefore, the sum \(0.6479953469975... + \frac{3}{4}\) is likely to also be irrational.
Thus, the only sum that results in a rational number is option C: 0.6 + 9/10.
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