Question

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

0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction

36−−√+221−−−√

Answer 1

To determine which of the sums results in a rational number based on the closure property, we need to analyze each option:

1. **\(0.6\overline{6} + \frac{9}{10}\)**:
- \(0.6\overline{6}\) is equivalent to \( \frac{2}{3} \) (a rational number).
- \(\frac{9}{10}\) is already a rational number.
- The sum \( \frac{2}{3} + \frac{9}{10} \) can be calculated as follows:
\[
\frac{2}{3} = \frac{20}{30} \quad \text{and} \quad \frac{9}{10} = \frac{27}{30}
\]
\[
\frac{20}{30} + \frac{27}{30} = \frac{47}{30}
\]
- So this sum is rational.

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

3. **\(0.643892553\ldots + \frac{3}{4}\)**:
- \(0.643892553\ldots\) is a non-repeating, non-terminating decimal and does not represent a rational number.
- \(\frac{3}{4}\) is a rational number.
- The sum of an irrational number and a rational number is irrational:
\[
0.643892553\ldots + \frac{3}{4} \text{ is irrational.}
\]

4. **\(\sqrt{36} + \sqrt{221}\)**:
- \(\sqrt{36} = 6\) (a rational number).
- \(\sqrt{221}\) is an irrational number.
- The sum of a rational number and an irrational number is irrational:
\[
6 + \sqrt{221} \text{ is irrational.}
\]

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

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