Question
According to the Closure Property, which of these sums results in a rational number? (1 point)
0 0.613892553 ..+ ¾
O 7+14
0 136 + v221
0 0.6 + 10
0 0.613892553 ..+ ¾
O 7+14
0 136 + v221
0 0.6 + 10
Answers
Answered by
GPT-4o mini
The Closure Property states that the sum (or product, etc.) of two rational numbers is always a rational number.
Let's analyze each option:
1. **0 + 0.613892553... + ¾**: The number \(0.613892553...\) is a decimal which, as far as we can tell, does not appear to be rational (it could be a non-terminating, non-repeating decimal). However, \(¾\) is rational. The overall sum could potentially be rational or irrational depending on the specifics of \(0.613892553...\), but we can't definitively say that the sum is rational without knowing if \(0.613892553...\) is rational.
2. **7 + 14**: Both \(7\) and \(14\) are rational numbers. The sum, \(7 + 14 = 21\), is also a rational number. This option is valid.
3. **136 + √221**: The number \(√221\) is not a rational number (since it cannot be expressed as a fraction of two integers). Therefore, the sum \(136 + √221\) is not guaranteed to be rational.
4. **0.6 + 10**: Both \(0.6\) and \(10\) are rational numbers. The sum \(0.6 + 10 = 10.6\) is also rational.
Based on the analysis, the options leading to a rational sum are:
- **7 + 14**
- **0.6 + 10**
Either option would be correct, but if you need to select only one, it's typically best to select **7 + 14** as it is straightforward and entirely composed of integers.
Let's analyze each option:
1. **0 + 0.613892553... + ¾**: The number \(0.613892553...\) is a decimal which, as far as we can tell, does not appear to be rational (it could be a non-terminating, non-repeating decimal). However, \(¾\) is rational. The overall sum could potentially be rational or irrational depending on the specifics of \(0.613892553...\), but we can't definitively say that the sum is rational without knowing if \(0.613892553...\) is rational.
2. **7 + 14**: Both \(7\) and \(14\) are rational numbers. The sum, \(7 + 14 = 21\), is also a rational number. This option is valid.
3. **136 + √221**: The number \(√221\) is not a rational number (since it cannot be expressed as a fraction of two integers). Therefore, the sum \(136 + √221\) is not guaranteed to be rational.
4. **0.6 + 10**: Both \(0.6\) and \(10\) are rational numbers. The sum \(0.6 + 10 = 10.6\) is also rational.
Based on the analysis, the options leading to a rational sum are:
- **7 + 14**
- **0.6 + 10**
Either option would be correct, but if you need to select only one, it's typically best to select **7 + 14** as it is straightforward and entirely composed of integers.