Question
According to the Closure Property, which of these sums results in a rational number? (1 point) 0.643892553...+ 3 4; sqrt(36) + sqrt(221); overline 0.6 + 9 1n; pi + sqrt(4)
Answers
Answered by
GPT-4o mini
To determine which of these sums results in a rational number, we need to evaluate each expression:
1. **0.643892553... + 3 4**: The number \(0.643892553...\) is a decimal (and thus a rational number), and \(3 4\) seems to be incorrectly formatted, but if interpreted as \(3 + 4\) (which equals \(7\), a rational number), the whole sum \(0.643892553... + 7\) results in a rational number.
2. **\(\sqrt{36} + \sqrt{221}\)**: \(\sqrt{36} = 6\), which is rational. However, \(\sqrt{221}\) is not a rational number (as 221 is not a perfect square), so the sum \(6 + \sqrt{221}\) is irrational.
3. **\(\overline{0.6} + 9 1n**: The term \(\overline{0.6}\) denotes the repeating decimal \(0.666...\), which is a rational number. The term \(9 1n\) appears to be incorrectly formatted. Assuming it means \(9 + 1\) (which equals \(10\), also a rational number), the sum \(0.666... + 10\) is rational.
4. **\(\pi + \sqrt{4}\)**: \(\sqrt{4} = 2\), which is rational. However, \(\pi\) is an irrational number, so \(2 + \pi\) is also irrational.
From the analysis, both **1** and **3** produce rational sums based on the assumptions made for \(3 4\) and \(9 1n\). Therefore, based on common interpretations of these formats, we can conclude that:
**The sums that result in rational numbers are \(0.643892553... + 7\) and \(0.666... + 10\)**. If only one answer can be selected and depending on the assumption of the interpretation of \(3 4\) and \(9 1n\), either could potentially be given as the final answer.
1. **0.643892553... + 3 4**: The number \(0.643892553...\) is a decimal (and thus a rational number), and \(3 4\) seems to be incorrectly formatted, but if interpreted as \(3 + 4\) (which equals \(7\), a rational number), the whole sum \(0.643892553... + 7\) results in a rational number.
2. **\(\sqrt{36} + \sqrt{221}\)**: \(\sqrt{36} = 6\), which is rational. However, \(\sqrt{221}\) is not a rational number (as 221 is not a perfect square), so the sum \(6 + \sqrt{221}\) is irrational.
3. **\(\overline{0.6} + 9 1n**: The term \(\overline{0.6}\) denotes the repeating decimal \(0.666...\), which is a rational number. The term \(9 1n\) appears to be incorrectly formatted. Assuming it means \(9 + 1\) (which equals \(10\), also a rational number), the sum \(0.666... + 10\) is rational.
4. **\(\pi + \sqrt{4}\)**: \(\sqrt{4} = 2\), which is rational. However, \(\pi\) is an irrational number, so \(2 + \pi\) is also irrational.
From the analysis, both **1** and **3** produce rational sums based on the assumptions made for \(3 4\) and \(9 1n\). Therefore, based on common interpretations of these formats, we can conclude that:
**The sums that result in rational numbers are \(0.643892553... + 7\) and \(0.666... + 10\)**. If only one answer can be selected and depending on the assumption of the interpretation of \(3 4\) and \(9 1n\), either could potentially be given as the final answer.