To determine which of the sums results in a rational number, let's evaluate each expression:
-
\(0.6\overline{6} + 9\frac{1}{10}\):
- \(0.6\overline{6}\) represents the repeating decimal \(0.666...\), which is rational and equals \(\frac{2}{3}\).
- \(9\frac{1}{10}\) is \(9 + 0.1 = 9.1\), which is also rational.
- Since both terms are rational, their sum (\(0.666... + 9.1\)) is rational.
-
\(0.643892553... + 340.643892553...\):
- \(0.643892553...\) is a non-repeating decimal (also called an irrational number since it cannot be expressed as a fraction).
- \(340.643892553...\) is also non-repeating and irrational.
- The sum of two irrational numbers can be rational or irrational, but mainly, in this form, it tends to be irrational.
-
\(\pi + 4\sqrt{4}\):
- \(\pi\) is a well-known irrational number.
- \(4\sqrt{4} = 4 \times 2 = 8\), which is rational.
- The sum of an irrational number and a rational number is irrational.
-
\(\sqrt{36} + 2\sqrt{1}\):
- \(\sqrt{36} = 6\), which is rational.
- \(2\sqrt{1} = 2\), which is also rational.
- The sum \(6 + 2 = 8\) is rational.
Based on this analysis, the sums that result in a rational number are:
- \(0.6\overline{6} + 9\frac{1}{10}\)
- \(\sqrt{36} + 2\sqrt{1}\)
In conclusion, both sums (1) and (4) result in rational numbers, but if you are only to select one depending on your question's context, the first one satisfies the closure property as well (if looking for an example).