Question

According to the Closure Property, which of these sums results in a rational number?(1 point) Responses 0.6¯¯¯¯¯¯+910 line segment 0 point 6 plus 9 tenths 36−−√+221−−−√ square root of 36 plus square root of 221 π+4–√ pi plus square root of 4 0.643892553...+34

Answer 1

To determine which of the given sums results in a rational number, we need to analyze each expression individually:

1. **\(0.6\overline{6} + \frac{9}{10}\)**:
- \(0.6\overline{6}\) represents the repeating decimal \(0.666...\), which is rational (\(\frac{2}{3}\)).
- \(\frac{9}{10}\) is also rational.
- The sum of two rational numbers is rational, so this expression is rational.

2. **\(\sqrt{36} + \sqrt{221}\)**:
- \(\sqrt{36} = 6\), which is rational.
- \(\sqrt{221}\) is not a perfect square and is therefore irrational.
- The sum of a rational number (6) and an irrational number (\(\sqrt{221}\)) is irrational.

3. **\(\pi + \sqrt{4}\)**:
- \(\sqrt{4} = 2\), which is rational.
- \(\pi\) is an irrational number.
- The sum of a rational number (2) and an irrational number (\(\pi\)) is irrational.

4. **\(0.643892553... + 34\)**:
- \(0.643892553...\) is a non-repeating decimal and thus irrational.
- \(34\) is rational.
- The sum of a rational number (34) and an irrational number (\(0.643892553...\)) is irrational.

Based on this analysis, the only expression that results in a rational number is:

**\(0.6\overline{6} + \frac{9}{10}\)**.