Question

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

12−−√+14
square root of 12 plus 1 fourth

5–√+7.89
square root of 5 plus 7 point 8 9

8.5¯¯¯+π
8 point repeating 5 plus pi

49−−√+2.33

Answer 1

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

1. **\(\sqrt{12} + \frac{1}{4}\)**:
- \(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\), which is an irrational number.
- Adding \(\frac{1}{4}\) (a rational number) to an irrational number (\(\sqrt{12}\)) will still yield an irrational result.

2. **\(\sqrt{5} + 7.89\)**:
- \(\sqrt{5}\) is an irrational number.
- \(7.89\) is a rational number.
- Adding an irrational number to a rational number yields an irrational result.

3. **\(8.\overline{5} + \pi\)**:
- \(8.\overline{5}\) is a rational number (it can be expressed as \(\frac{17}{2}\)).
- \(\pi\) is an irrational number.
- Adding a rational number to an irrational number results in an irrational number.

4. **\(\sqrt{49} + 2.33\)**:
- \(\sqrt{49} = 7\), which is a rational number.
- \(2.33\) is also a rational number.
- Adding two rational numbers results in a rational number.

Among the given sums, the only one that results in a rational number is:

**\(\sqrt{49} + 2.33\)**.