Question
According to the Closure Property, 14+2–√14+2 would result in what type of number?(1 point) Responses Integer Integer Zero Zero Rational Rational Irrational Irrational Check Answer Question 2 According to the Closure Property, which of these differences would result in a rational number?(1 point) Responses 64−−√−7–√64−7square root of 64 minus square root of 7 49−−√−1449−14square root of 49 minus 1 fourth 5–√−3.25−3.2square root of 5 minus 3 point 2 6.2¯¯¯−π6.2¯−π6 point repeating 2 minus pi
Answers
GPT-4o mini
Let's go through the questions one by one.
### Question 1
You are asked to determine the type of number resulting from the expression \( 14 + 2 - \sqrt{14 + 2} \).
1. **Calculate \( 14 + 2 \):**
\[
14 + 2 = 16
\]
2. **Calculate \( \sqrt{16} \):**
\[
\sqrt{16} = 4
\]
3. **Now substitute back into the expression:**
\[
16 - 4 = 12
\]
Now, we identify the type of number:
- \( 12 \) is an **integer** and also a **rational** number (since all integers are rational numbers).
Based solely on the options, the most straightforward answer would be **Integer**.
### Question 2
You need to identify which of the given differences results in a rational number:
1. **Option 1: \( \sqrt{64} - \sqrt{7} \)**
- \( \sqrt{64} = 8 \), so \( 8 - \sqrt{7} \). Since \( \sqrt{7} \) is irrational, this result is also irrational.
2. **Option 2: \( \sqrt{49} - \frac{1}{4} \)**
- \( \sqrt{49} = 7 \), so \( 7 - \frac{1}{4} = \frac{28}{4} - \frac{1}{4} = \frac{27}{4} \), which is rational.
3. **Option 3: \( \sqrt{5} - 3.2 \)**
- \( \sqrt{5} \) is irrational, and subtracting a rational number (3.2) from an irrational number results in an irrational number.
4. **Option 4: \( 6.2 - \pi \)**
- \( 6.2 \) is rational, while \( \pi \) is irrational. Thus, their difference is also irrational.
Based on the evaluation, the only option that results in a rational number is **Option 2: \( \sqrt{49} - \frac{1}{4} \)**.
### Summary
- **Question 1 Answer:** Integer
- **Question 2 Answer:** \( \sqrt{49} - \frac{1}{4} \) (which simplifies to a rational number).
### Question 1
You are asked to determine the type of number resulting from the expression \( 14 + 2 - \sqrt{14 + 2} \).
1. **Calculate \( 14 + 2 \):**
\[
14 + 2 = 16
\]
2. **Calculate \( \sqrt{16} \):**
\[
\sqrt{16} = 4
\]
3. **Now substitute back into the expression:**
\[
16 - 4 = 12
\]
Now, we identify the type of number:
- \( 12 \) is an **integer** and also a **rational** number (since all integers are rational numbers).
Based solely on the options, the most straightforward answer would be **Integer**.
### Question 2
You need to identify which of the given differences results in a rational number:
1. **Option 1: \( \sqrt{64} - \sqrt{7} \)**
- \( \sqrt{64} = 8 \), so \( 8 - \sqrt{7} \). Since \( \sqrt{7} \) is irrational, this result is also irrational.
2. **Option 2: \( \sqrt{49} - \frac{1}{4} \)**
- \( \sqrt{49} = 7 \), so \( 7 - \frac{1}{4} = \frac{28}{4} - \frac{1}{4} = \frac{27}{4} \), which is rational.
3. **Option 3: \( \sqrt{5} - 3.2 \)**
- \( \sqrt{5} \) is irrational, and subtracting a rational number (3.2) from an irrational number results in an irrational number.
4. **Option 4: \( 6.2 - \pi \)**
- \( 6.2 \) is rational, while \( \pi \) is irrational. Thus, their difference is also irrational.
Based on the evaluation, the only option that results in a rational number is **Option 2: \( \sqrt{49} - \frac{1}{4} \)**.
### Summary
- **Question 1 Answer:** Integer
- **Question 2 Answer:** \( \sqrt{49} - \frac{1}{4} \) (which simplifies to a rational number).