To determine which of the given sums results in a rational number under the Closure Property, let's analyze each case step-by-step:
1. Sum: [tex]\(\pi + \sqrt{4}\)[/tex]
- [tex]\(\pi\)[/tex] is known to be an irrational number.
- [tex]\(\sqrt{4}\)[/tex] simplifies to 2, which is a rational number.
The sum of an irrational number ([tex]\(\pi\)[/tex]) and a rational number (2) is generally irrational.
Hence, [tex]\(\pi + \sqrt{4}\)[/tex] is not rational.
2. Sum: [tex]\(\sqrt{36} + \sqrt{221}\)[/tex]
- [tex]\(\sqrt{36}\)[/tex] simplifies to 6, which is a rational number.
- [tex]\(\sqrt{221}\)[/tex] is an irrational number.
The sum of a rational number (6) and an irrational number ([tex]\(\sqrt{221}\)[/tex]) is generally irrational.
Hence, [tex]\(\sqrt{36} + \sqrt{221}\)[/tex] is not rational.
3. Sum: [tex]\(0.643892553\ldots + \frac{3}{4}\)[/tex]
- [tex]\(0.643892553 \ldots\)[/tex] represents a decimal number that we assume to be rational.
- [tex]\(\frac{3}{4}\)[/tex] is a rational number.
The sum of two rational numbers is always rational.
Hence, [tex]\(0.643892553\ldots + \frac{3}{4}\)[/tex] is rational.
4. Sum: [tex]\(0.\overline{6} + \frac{9}{10}\)[/tex]
- [tex]\(0.\overline{6}\)[/tex] is a repeating decimal, which can be converted to [tex]\(\frac{2}{3}\)[/tex], a rational number.
- [tex]\(\frac{9}{10}\)[/tex] is a rational number.
The sum of two rational numbers is always rational.
Hence, [tex]\(0.\overline{6} + \frac{9}{10}\)[/tex] is rational.
In conclusion, the sums that result in rational numbers are:
- [tex]\(0.643892553 \ldots + \frac{3}{4}\)[/tex]
- [tex]\(0.\overline{6} + \frac{9}{10}\)[/tex]
