Question

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

√ 36+√221

π+4–√

0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction


0.643892553…+3/4

Answer 1

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

1. **\( \sqrt{36} + \sqrt{221} \)**:
- \( \sqrt{36} = 6 \) (a rational number).
- \( \sqrt{221} \) is an irrational number (since it cannot be expressed as a ratio of integers).
- Sum: \( 6 + \sqrt{221} \) is irrational.

2. **\( \pi + 4 - \sqrt{} \)**:
- \( \pi \) is an irrational number, and \( \sqrt{} \) (presumably you meant a square root of a non-perfect square) is also irrational.
- Sum: Any combination of rational and irrational numbers results in an irrational number. Therefore, this sum is irrational.

3. **\( 0.\overline{6} + \frac{9}{10} \)**:
- \( 0.\overline{6} \) (which is \( \frac{2}{3} \) when converted to a fraction) is a rational number.
- \( \frac{9}{10} \) is also a rational number.
- Sum: \( \frac{2}{3} + \frac{9}{10} \) is a rational number (both parts are rational).

4. **\( 0.643892553… + \frac{3}{4} \)**:
- \( 0.643892553… \) is a non-repeating, non-terminating decimal which typically signifies it is irrational.
- \( \frac{3}{4} \) is a rational number.
- Sum: An irrational number plus a rational number results in an irrational number.

Based on this analysis, the only sum that results in a rational number is:

**0.\overline{6} + \frac{9}{10}**.