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 results in a rational number, let's evaluate each expression:

1. \( 5 - \sqrt{2} \): This expression involves the subtraction of the irrational number \(\sqrt{2}\) from the rational number 5. The result is irrational.

2. \( \sqrt{36} - \frac{2}{3} \): Here, \(\sqrt{36} = 6\) is a rational number, and \(\frac{2}{3}\) is also rational. The difference \(6 - \frac{2}{3} = \frac{18}{3} - \frac{2}{3} = \frac{16}{3}\) is rational.

3. \( \pi - 1 \): Since \(\pi\) is an irrational number and 1 is rational, their difference is also irrational.

4. \( 6.2 - \sqrt{3} \): Here, 6.2 is rational, and \(\sqrt{3}\) is irrational, so \(6.2 - \sqrt{3}\) is irrational.

Out of these options, only the difference \( \sqrt{36} - \frac{2}{3} \) results in a rational number. Thus, the answer is:

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