Question

According to the Closure Property, which of these sums results in a rational number?(1 point)
Responses

0.6¯¯¯¯+910
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction
36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root
0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction
π+4–√

Answer 1

To determine which of the sums results in a rational number, let's analyze each option:

1. **0.6¯ + 9/10**:
- \(0.6̅\) represents the repeating decimal \(0.666...\), which is equal to \( \frac{2}{3} \).
- So, \( \frac{2}{3} + \frac{9}{10} \) needs to be calculated:
- To add these fractions, find a common denominator, which is 30.
- \( \frac{2}{3} = \frac{20}{30} \) and \( \frac{9}{10} = \frac{27}{30} \).
- \( \frac{20}{30} + \frac{27}{30} = \frac{47}{30} \), which is rational.

2. **√36 + √221**:
- √36 = 6 (which is rational) and √221 is not a perfect square, so it is irrational.
- The sum \( 6 + \sqrt{221} \) is irrational.

3. **0.643892553... + 3/4**:
- The decimal is a non-terminating, non-repeating decimal and treated as irrational.
- Adding \(3/4\) (which is rational) to an irrational number results in an irrational number.

4. **π + √4**:
- π is irrational and √4 = 2 (which is rational).
- Again, the sum of a rational and an irrational number is irrational.

Based on the analysis, the first option **0.6̅ + 9/10** results in a rational number.

### Conclusion:
The sum that results in a rational number is:
**0.6̅ + 9/10**.