To determine which of the sums results in a rational number, we need to evaluate each expression:
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\(0.643892553... + 34\): The number \(0.643892553...\) is a non-repeating decimal and thus an irrational number. The sum of an irrational number and a rational number (34) is irrational, so this expression does not yield a rational number.
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\(\pi + 4 - \sqrt{4}\): Here, \(\pi\) is an irrational number, \(4\) is rational, and \(\sqrt{4} = 2\) is rational. Therefore, the expression simplifies to \(\pi + 4 - 2 = \pi + 2\), which is still irrational because the sum of an irrational number and a rational number is irrational. Thus, this does not yield a rational number.
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\(\sqrt{36} + \sqrt{221}\): The square root of \(36\) is \(6\), which is rational, but \(\sqrt{221}\) is irrational (since \(221\) is not a perfect square). The sum \(6 + \sqrt{221}\) will be irrational (as the sum of a rational and an irrational number is irrational). So this does not yield a rational number.
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\(0.6\overline{6} + \frac{9}{10}\): The number \(0.6\overline{6}\) is a repeating decimal, which is rational (it can be expressed as \( \frac{2}{3} \)). The fraction \(\frac{9}{10}\) is also rational. The sum of two rational numbers is rational. Thus, \(0.6\overline{6} + \frac{9}{10}\) results in a rational number.
Therefore, the correct answer is:
\(0.6\overline{6} + \frac{9}{10}\) results in a rational number.