Question
According to the Closure Property, which of these sums results in a rational number?(1 point)
Responses
π+4–√
pi plus Start Root 4 End Root
0.6¯¯¯¯+910
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction
0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction
36−−√+221−−−√
Responses
π+4–√
pi plus Start Root 4 End Root
0.6¯¯¯¯+910
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction
0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction
36−−√+221−−−√
Answers
Answered by
GPT-4o mini
To determine which of the given sums results in a rational number, we need to analyze each option based on the properties of rational and irrational numbers.
1. **π + √4**
- Here, \(\pi\) is an irrational number and \(\sqrt{4} = 2\), which is a rational number. The sum of a rational number and an irrational number is irrational.
- Therefore, π + √4 is **irrational**.
2. **0.6̅ + 9/10**
- The decimal \(0.6̅\) (which represents \(0.666...\)) is a rational number (it can be expressed as the fraction \(2/3\)). The fraction \(9/10\) is also a rational number.
- The sum of two rational numbers is rational.
- Therefore, \(0.6̅ + 9/10\) is **rational**.
3. **0.643892553… + 3/4**
- The number \(0.643892553...\) is a non-repeating decimal and is classified as an irrational number. The fraction \(3/4\) is a rational number.
- The sum of a rational number and an irrational number is irrational.
- Therefore, \(0.643892553… + 3/4\) is **irrational**.
4. **√36 + √221**
- Here, \(\sqrt{36} = 6\) is a rational number, but \(\sqrt{221}\) is irrational (since 221 is not a perfect square).
- The sum of a rational number and an irrational number is irrational.
- Therefore, \(√36 + √221\) is **irrational**.
**Conclusion:**
The only sum that results in a rational number is **0.6̅ + 9/10**.
1. **π + √4**
- Here, \(\pi\) is an irrational number and \(\sqrt{4} = 2\), which is a rational number. The sum of a rational number and an irrational number is irrational.
- Therefore, π + √4 is **irrational**.
2. **0.6̅ + 9/10**
- The decimal \(0.6̅\) (which represents \(0.666...\)) is a rational number (it can be expressed as the fraction \(2/3\)). The fraction \(9/10\) is also a rational number.
- The sum of two rational numbers is rational.
- Therefore, \(0.6̅ + 9/10\) is **rational**.
3. **0.643892553… + 3/4**
- The number \(0.643892553...\) is a non-repeating decimal and is classified as an irrational number. The fraction \(3/4\) is a rational number.
- The sum of a rational number and an irrational number is irrational.
- Therefore, \(0.643892553… + 3/4\) is **irrational**.
4. **√36 + √221**
- Here, \(\sqrt{36} = 6\) is a rational number, but \(\sqrt{221}\) is irrational (since 221 is not a perfect square).
- The sum of a rational number and an irrational number is irrational.
- Therefore, \(√36 + √221\) is **irrational**.
**Conclusion:**
The only sum that results in a rational number is **0.6̅ + 9/10**.