Question

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

Answer 1

To determine which sum results in a rational number, we need to evaluate each expression based on the concept of the closure property for rational numbers (which states that the sum of two rational numbers is also a rational number).

1. **\(\sqrt{36} + \sqrt{221}\)**:
- \(\sqrt{36} = 6\), which is rational.
- \(\sqrt{221}\) is not a perfect square and is an irrational number.
- Therefore, \(6 + \sqrt{221}\) is irrational.

2. **\(0.6\overline{6} + \frac{9}{10}\)**:
- \(0.6\overline{6}\) (which is \(0.666...\)) can be expressed as \(\frac{2}{3}\), which is a rational number.
- \(\frac{9}{10}\) is also a rational number.
- The sum \(\frac{2}{3} + \frac{9}{10}\) is rational (it can be computed as \(\frac{20}{30} + \frac{27}{30} = \frac{47}{30}\)).

3. **\(\pi + \sqrt{4}\)**:
- \(\sqrt{4} = 2\) is rational.
- \(\pi\) is irrational.
- Therefore, \(\pi + 2\) is irrational.

4. **\(0.643892553... + 3\frac{4}{10}\)**:
- \(0.643892553...\) is an example of a non-repeating decimal, which is typically considered irrational.
- \(3\frac{4}{10} = 3.4\) is rational.
- Thus, the sum of an irrational number and a rational number is irrational.

Among the sums provided, the only sum that results in a rational number is:

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