Let's analyze each of the given sums to determine which one results in a rational number:
1. Sum [tex]\( \sqrt{36} + \sqrt{221} \)[/tex]
- The square root of 36 is [tex]\(\sqrt{36} = 6\)[/tex], which is a rational number.
- The square root of 221, [tex]\(\sqrt{221}\)[/tex], is an irrational number since 221 is not a perfect square.
- A sum involving a rational number (6) and an irrational number ([tex]\(\sqrt{221}\)[/tex]) remains irrational.
Therefore, [tex]\(\sqrt{36} + \sqrt{221}\)[/tex] does not result in a rational number.
2. Sum [tex]\( \pi + \sqrt{4} \)[/tex]
- The value of [tex]\(\pi\)[/tex] is an irrational number.
- The square root of 4 is [tex]\(\sqrt{4} = 2\)[/tex], which is a rational number.
- Adding an irrational number ([tex]\(\pi\)[/tex]) to a rational number (2) results in an irrational number.
Therefore, [tex]\(\pi + \sqrt{4}\)[/tex] does not result in a rational number.
3. Sum [tex]\( 0.\overline{6} + \frac{9}{10} \)[/tex]
- The repeating decimal [tex]\(0.\overline{6}\)[/tex] can be expressed as the fraction [tex]\(\frac{2}{3}\)[/tex], which is rational.
- The fraction [tex]\(\frac{9}{10}\)[/tex] is already a rational number.
- Adding these two rational numbers:
[tex]\[
0.\overline{6} + \frac{9}{10} = \frac{2}{3} + \frac{9}{10}
\][/tex]
Finding a common denominator (which is 30):
[tex]\[
\frac{2}{3} = \frac{20}{30},\quad \frac{9}{10} = \frac{27}{30}
\][/tex]
Adding these fractions:
[tex]\[
\frac{20}{30} + \frac{27}{30} = \frac{47}{30}
\][/tex]
The fraction [tex]\(\frac{47}{30}\)[/tex] is rational.
Hence, [tex]\(0.\overline{6} + \frac{9}{10}\)[/tex] results in a rational number.
4. Sum [tex]\( 0.643892553\ldots + \frac{3}{4} \)[/tex]
- The number [tex]\(0.643892553\ldots\)[/tex] is an approximation and appears to be an irrational number.
- The fraction [tex]\(\frac{3}{4}\)[/tex] is a rational number.
- Adding an irrational number to a rational number results in an irrational number.
Therefore, [tex]\(0.643892553\ldots + \frac{3}{4}\)[/tex] does not result in a rational number.
In conclusion, the sum that results in a rational number is:
[tex]\[ 0.\overline{6} + \frac{9}{10} \][/tex]
