To determine which of the given sums is a rational number, let's evaluate each one:
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Pi + square root 4: \[ \pi + \sqrt{4} = \pi + 2 \] Since \(\pi\) is an irrational number, the sum \(\pi + 2\) is also irrational. Therefore, this is not a rational number.
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0.643892553… + 3/4: The number \(0.643892553…\) is not a simple repeating decimal or a rational number dressable in form. So we cannot directly conclude if this sum is rational without knowing more about the decimal expansion. In many cases, it's likely also irrational, but we cannot assume it is rational.
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0.6 (repeating) + 9/10: Here, \(0.6\) repeating is written as: \[ 0.6666\ldots = \frac{2}{3} \] Now combining it with \(9/10\): \[ \frac{2}{3} + \frac{9}{10} = \frac{20}{30} + \frac{27}{30} = \frac{47}{30} \] Since \(\frac{47}{30}\) is a rational number, this sum is rational.
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Square root 36 + square root 221: Evaluating this gives: \[ \sqrt{36} + \sqrt{221} = 6 + \sqrt{221} \] Since \(\sqrt{221}\) is an irrational number, the sum \(6 + \sqrt{221}\) is also irrational.
Thus, the only sum that results in a rational number is 0.6 repeated + 9/10.
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