To determine which of the given sums results in a rational number, we need to evaluate each sum step by step and check if the result is a rational number.
1. Sum 1: [tex]\(\sqrt{36} + \sqrt{221}\)[/tex]
[tex]\[
\sqrt{36} = 6
\][/tex]
[tex]\[
\sqrt{221} \approx 14.866068747318506
\][/tex]
Adding these results gives:
[tex]\[
6 + 14.866068747318506 \approx 20.866068747318508
\][/tex]
This is not an integer, and therefore not a rational number.
2. Sum 2: [tex]\(0 . \overline{6} + \frac{9}{10}\)[/tex]
[tex]\[
0 . \overline{6} = \frac{2}{3}
\][/tex]
[tex]\[
\frac{9}{10} = 0.9
\][/tex]
Adding these results gives:
[tex]\[
\frac{2}{3} + 0.9 \approx 1.5666666666666667
\][/tex]
This is not an integer, and therefore not a rational number.
3. Sum 3: [tex]\(\pi + \sqrt{4}\)[/tex]
[tex]\[
\pi \approx 3.141592653589793
\][/tex]
[tex]\[
\sqrt{4} = 2
\][/tex]
Adding these results gives:
[tex]\[
3.141592653589793 + 2 \approx 5.141592653589793
\][/tex]
This is not an integer, and therefore not a rational number.
4. Sum 4: [tex]\(0.643892553\ldots + \frac{3}{4}\)[/tex]
[tex]\[
0.643892553 \approx 0.643892553
\][/tex]
[tex]\[
\frac{3}{4} = 0.75
\][/tex]
Adding these results gives:
[tex]\[
0.643892553 + 0.75 \approx 1.393892553
\][/tex]
This is not an integer, and therefore not a rational number.
Based on the evaluations, none of the sums given result in a rational number. Each sum results in an irrational number. Therefore, no sum satisfies the Closure Property for rational numbers.
Answer: None of these sums results in a rational number.
