Question

Which of these is a rational number that comes between √7 and √8?

answers:

5/6

19/7

3/4

6/3

Answer 1

are you sure that is the right answer?

Answer 2

To find which of these options is a rational number between \( \sqrt{7} \) and \( \sqrt{8} \), we first need to approximate the square roots.

Calculating \( \sqrt{7} \) and \( \sqrt{8} \):
- \( \sqrt{7} \approx 2.64575 \)
- \( \sqrt{8} = 2\sqrt{2} \approx 2.82843 \)

Now, we can check which of the given rational numbers lies between \( \sqrt{7} \) and \( \sqrt{8} \).

1. **Check \( \frac{5}{6} \):**
\[
\frac{5}{6} \approx 0.8333 \quad (\text{not between } 2.64575 \text{ and } 2.82843)
\]

2. **Check \( \frac{19}{7} \):**
\[
\frac{19}{7} \approx 2.7143 \quad (\text{between } 2.64575 \text{ and } 2.82843)
\]

3. **Check \( \frac{3}{4} \):**
\[
\frac{3}{4} = 0.75 \quad (\text{not between } 2.64575 \text{ and } 2.82843)
\]

4. **Check \( \frac{6}{3} \):**
\[
\frac{6}{3} = 2 \quad (\text{not between } 2.64575 \text{ and } 2.82843)
\]

Out of all the options, \( \frac{19}{7} \) is the only rational number that lies between \( \sqrt{7} \) and \( \sqrt{8} \).

Therefore, the answer is \( \frac{19}{7} \).

Answer 3

Let's double-check the calculations step-by-step to confirm whether \( \frac{19}{7} \) really lies between \( \sqrt{7} \) and \( \sqrt{8} \).

1. **Confirm values of \( \sqrt{7} \) and \( \sqrt{8} \)**:
- \( \sqrt{7} \approx 2.64575 \)
- \( \sqrt{8} = 2\sqrt{2} \approx 2.82843 \)

2. **Calculate \( \frac{19}{7} \)**:
\[
\frac{19}{7} \approx 2.714285714285714 \quad (\text{which is roughly } 2.7143)
\]

Now, comparing \( \frac{19}{7} \) to \( \sqrt{7} \) and \( \sqrt{8} \):
- Since \( 2.64575 < 2.7143 < 2.82843 \), we can conclude that \( \frac{19}{7} \) is indeed between \( \sqrt{7} \) and \( \sqrt{8} \).

Thus, my previous answer stands verified; \( \frac{19}{7} \) is a rational number that lies between \( \sqrt{7} \) and \( \sqrt{8} \). If you have any further questions or need clarification, feel free to ask!