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 0.643892553…+34 0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction 36−−√+221−−−√ Start Root 36 End Root plus Start Root 221 End Root 0.6¯¯¯¯+910

Answer 1

To determine which sums result in a rational number, we need to analyze each expression individually:

1. **π + 4**:
- π is an irrational number, and 4 is a rational number. The sum of a rational number and an irrational number is irrational.
- Result: **Irrational**

2. **–√(4)**:
- √(4) = 2, which is rational. Therefore, -√(4) = -2, which is also rational.
- Result: **Rational**

3. **0.643892553… + 34**:
- The number 0.643892553… appears to be a non-repeating, non-terminating decimal, so it is likely to be irrational. Adding a rational number (34) to an irrational number results in an irrational number.
- Result: **Irrational**

4. **0.643892553 dot dot dot + 3/4**:
- Similar to the previous case, if 0.643892553… is irrational, adding the rational number 3/4 results in an irrational number.
- Result: **Irrational**

5. **√(36) + √(221)**:
- √(36) = 6 (which is rational) and √(221) is irrational (since 221 is not a perfect square). The sum of a rational number (6) and an irrational number (√(221)) is irrational.
- Result: **Irrational**

6. **0.6̅ + 910**:
- 0.6̅ represents the repeating decimal 0.666... which is rational (it can be expressed as 2/3). Adding the rational number 910 gives another rational number.
- Result: **Rational**

Based on this analysis:

- The sums that result in a rational number are:
- **–√(4)** (which results in -2)
- **0.6̅ + 910** (which results in a rational number)

Thus, the answers you are looking for are **–√(4)** and **0.6̅ + 910** as these result in rational numbers.