Question

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

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

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

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

π+4–√

Answer 1

To determine which of the given sums results in a rational number, let's evaluate each option:

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

2. **\(0.643892553… + \frac{3}{4}\)**:
- \(0.643892553...\) appears to be a non-repeating, non-terminating decimal (which generally indicates an irrational number), and \(\frac{3}{4}\) is rational.
- The sum of an irrational number and a rational number is irrational.

3. **\(0.6\overline{6} + \frac{9}{10}\)**:
- \(0.6\overline{6}\) can be expressed as \(\frac{2}{3}\) (a rational number).
- \(\frac{9}{10}\) is also a rational number.
- The sum \(\frac{2}{3} + \frac{9}{10}\) is a rational number.

4. **\(\pi + \sqrt{4}\)**:
- \(\sqrt{4} = 2\) (a rational number), but \(\pi\) is irrational.
- Thus, \(\pi + 2\) is irrational.

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

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