Asked by Landon Sharp
According to the Closure Property, which of these differences would result in a rational number?(1 point)
Responses
5–√−2–√
5
−
2
square root of 5 minus square root of 2
36−−√−23
36
−
2
3
square root of 36 minus 2 thirds
π−1
𝜋
−
1
pi minus 1
6.2¯¯¯−3–√
Responses
5–√−2–√
5
−
2
square root of 5 minus square root of 2
36−−√−23
36
−
2
3
square root of 36 minus 2 thirds
π−1
𝜋
−
1
pi minus 1
6.2¯¯¯−3–√
Answers
Answered by
GPT-4o mini
The Closure Property of rational numbers states that the result of an operation (such as addition, subtraction, multiplication, or division) on rational numbers will also be a rational number. To determine which of the given differences results in a rational number, we can analyze each option:
1. **\( 5 - \sqrt{2} \)**: Here, \( 5 \) is rational, but \( \sqrt{2} \) is irrational. The difference of a rational number and an irrational number is irrational. So, this is not a rational number.
2. **\( \sqrt{36} - \frac{2}{3} \)**: \( \sqrt{36} = 6 \), which is rational, and \( \frac{2}{3} \) is also rational. The difference of two rational numbers is rational. So, this is a rational number.
3. **\( \pi - 1 \)**: \( \pi \) is irrational, and \( 1 \) is rational. The difference of a rational number and an irrational number is irrational. So, this is not a rational number.
4. **\( 6.2\overline{2} - \sqrt{3} \)**: \( 6.2\overline{2} \) (which equals \( 6.2222... \)) is rational (it can be expressed as a fraction), while \( \sqrt{3} \) is irrational. The difference of a rational number and an irrational number is irrational. So, this is not a rational number.
From the analysis above, the only difference that results in a rational number is:
**\( \sqrt{36} - \frac{2}{3} \) (which is \( 6 - \frac{2}{3} \)).**
1. **\( 5 - \sqrt{2} \)**: Here, \( 5 \) is rational, but \( \sqrt{2} \) is irrational. The difference of a rational number and an irrational number is irrational. So, this is not a rational number.
2. **\( \sqrt{36} - \frac{2}{3} \)**: \( \sqrt{36} = 6 \), which is rational, and \( \frac{2}{3} \) is also rational. The difference of two rational numbers is rational. So, this is a rational number.
3. **\( \pi - 1 \)**: \( \pi \) is irrational, and \( 1 \) is rational. The difference of a rational number and an irrational number is irrational. So, this is not a rational number.
4. **\( 6.2\overline{2} - \sqrt{3} \)**: \( 6.2\overline{2} \) (which equals \( 6.2222... \)) is rational (it can be expressed as a fraction), while \( \sqrt{3} \) is irrational. The difference of a rational number and an irrational number is irrational. So, this is not a rational number.
From the analysis above, the only difference that results in a rational number is:
**\( \sqrt{36} - \frac{2}{3} \) (which is \( 6 - \frac{2}{3} \)).**