https://r2.erweima.ai/mathbot/image/1724426583052z508vl6v.jpg

To solve the problem, we need to determine which of the given expressions have values between 7 and 8. Let's evaluate each expression:

1. **\(\sqrt{48}\)**:
   \[
   \sqrt{48} \approx 6.93
   \]
   This value is **not** between 7 and 8.

2. **\(\sqrt{56}\)**:
   \[
   \sqrt{56} \approx 7.48
   \]
   This value **is** between 7 and 8.

3. **\(2\pi\)**:
   \[
   2\pi \approx 6.28
   \]
   This value is **not** between 7 and 8.

4. **\(\pi + 4\)**:
   \[
   \pi \approx 3.14, \quad \pi + 4 \approx 7.14
   \]
   This value **is** between 7 and 8.

5. **\(\sqrt{62}\)**:
   \[
   \sqrt{62} \approx 7.87
   \]
   This value **is** between 7 and 8.

### Conclusion
The three expressions that have values between 7 and 8 are:
- \(\sqrt{56}\)
- \(\pi + 4\)
- \(\sqrt{62}\)

Do you want any more details or have any questions?

Here are 5 related questions to expand on this topic:
1. What is the exact value of \(\pi\) and why is it often approximated?
2. How can you estimate square roots without a calculator?
3. What is the value of \(\pi^2\), and how does it compare to \(2\pi\)?
4. How do you derive the value of \(\sqrt{56}\) using prime factorization?
5. Can you find a value between \(2\pi\) and \(\sqrt{62}\)?

**Tip:** Remember that \(\pi\) is an irrational number, so its decimal expansion goes on forever without repeating, which is why we often use approximations like 3.14 or 22/7.

Explore the evaluation of mathematical expressions involving square roots, pi, and approximations between 7 and 8. Learn how to approximate square roots and understand the value of pi in different contexts. Suitable for advanced high school students and those interested in numerical approximations.

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