Question

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

5–√−2–√
square root of 5 minus square root of 2

36−−√−23
square root of 36 minus 2 thirds

π−1
pi minus 1

6.2¯¯¯−3–√

Answer 1

To determine which of the given differences would result in a rational number, we need to analyze each expression individually.

1. **\(5 - \sqrt{2}\)**: This involves rational number 5 and the irrational number \( \sqrt{2} \). The difference will be irrational because the subtraction of a rational and an irrational number is always irrational.

2. **\( \sqrt{36} - \frac{2}{3} \)**: The first term, \( \sqrt{36} \), is 6, which is a rational number. Subtracting another rational number \( \frac{2}{3} \) from it results in \( 6 - \frac{2}{3} = \frac{18}{3} - \frac{2}{3} = \frac{16}{3} \), which is a rational number.

3. **\( \pi - 1 \)**: Here, \( \pi \) is an irrational number, and 1 is a rational number. The difference \( \pi - 1 \) is still irrational.

4. **\( 6.2\overline{6} - \sqrt{3} \)**: The number \( 6.2\overline{6} \) is a rational number (it is equivalent to \( \frac{188}{30} \)). \( \sqrt{3} \) is an irrational number, so the difference would also be irrational.

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

**\( \sqrt{36} - \frac{2}{3} \)**.