Question

According to the Closure Property, which of these sums results in a rational number?(1 point)
Responses

0.6¯¯¯¯+910
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction

π+4–√
pi plus Start Root 4 End Root

36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root

0.643892553…+34

Answer 1

To determine which of the given sums results in a rational number, we will analyze each option one by one.

1. **\(0.6\overline{6} + \frac{9}{10}\)**
- \(0.6\overline{6}\) is a repeating decimal which can be expressed as the fraction \(\frac{2}{3}\).
- Therefore, \(0.6\overline{6} + \frac{9}{10} = \frac{2}{3} + \frac{9}{10}\).
- To add these fractions, we need a common denominator: the least common multiple of 3 and 10 is 30.
\[
\frac{2}{3} = \frac{20}{30}, \quad \frac{9}{10} = \frac{27}{30} \implies \frac{20}{30} + \frac{27}{30} = \frac{47}{30} \text{ (rational)}
\]

2. **\(\pi + \sqrt{4}\)**
- \(\sqrt{4}\) equals 2, which is rational.
- However, \(\pi\) is an irrational number. The sum of a rational number and an irrational number is irrational.
\[
\pi + 2 \text{ is irrational.}
\]

3. **\(\sqrt{36} + \sqrt{221}\)**
- \(\sqrt{36}\) equals 6, which is rational.
- \(\sqrt{221}\) is irrational, being the square root of a non-square integer. The sum of a rational number and an irrational number is also irrational.
\[
6 + \sqrt{221} \text{ is irrational.}
\]

4. **\(0.643892553... + 34\)**
- The number \(0.643892553...\) appears to be a non-repeating, non-terminating decimal, which indicates that it is irrational.
- Thus, the sum of an irrational number (0.643892553...) and a rational number (34) is also irrational.
\[
0.643892553... + 34 \text{ is irrational.}
\]

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

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