Question

According to the Closure Property, which of these sums results in a rational number? Responses 0.643892553…+34 0 point 6 4 3 8 9 2 5 5 3 dot dot dot plus 3 fourths - no response given π+4–√ pi plus square root of 4 - no response given 36−−√+221−−−√ square root of 36 plus square root of 221 - no response given 0.6¯¯¯¯+910

Answers

Answered by GPT-4o mini
To determine which of the given sums results in a rational number, we need to analyze each one based on the Closure Property of rational numbers, which states that the sum of any two rational numbers is also a rational number.

1. **0.643892553… + 34**: The number 0.643892553… is a decimal with a repeating non-terminating nature. However, as a repeating decimal, it can be expressed as a rational number. Therefore, 0.643892553… + 34 is rational, since both terms are rational numbers.

2. **π + 4 - √**: Here, π (pi) is an irrational number. Since irrational numbers do not produce a rational number when added to a rational number (4 is rational), the sum π + 4 is irrational. The same applies for the term -√ (presumably referring to some square root that also can be irrational). Therefore, this sum results in an irrational number.

3. **√36 + √221**: The square root of 36 is 6 (which is rational). However, the square root of 221 is not a perfect square and thus is an irrational number. The sum of a rational number (6) and an irrational number (√221) is irrational. So, this sum is also irrational.

4. **0.6¯¯¯¯ + 910**: The notation 0.6¯¯¯¯ indicates the repeating decimal 0.6666…, which is equal to the rational number 2/3. If we take 2/3 and add it to 910 (a rational number), the result is rational. Hence, this sum is also rational.

In summary:
- **0.643892553… + 34**: Rational
- **π + 4 - √**: Irrational
- **√36 + √221**: Irrational
- **0.6 + 910**: Rational

Thus, the sums resulting in rational numbers are **0.643892553… + 34** and **0.6 + 910**.

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