Question

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

Responses

5/6

19/7

3/4

6/3

Answer 1

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

- \( \sqrt{7} \approx 2.6457513110645906 \)
- \( \sqrt{8} = 2\sqrt{2} \approx 2.8284271247461903 \)

Now we can observe that both values are between 2.6 and 2.8. We need to find a rational number between these two square roots.

Let's check the given options one by one to see if any fall between 2.6457513110645906 and 2.8284271247461903:

1. **5/6:**
\[
\frac{5}{6} \approx 0.8333 \quad \text{(not between 2.645 and 2.828)}
\]

2. **19/7:**
\[
\frac{19}{7} \approx 2.714285714285714 \quad \text{(this is between 2.645 and 2.828)}
\]

3. **3/4:**
\[
\frac{3}{4} = 0.75 \quad \text{(not between 2.645 and 2.828)}
\]

4. **6/3:**
\[
\frac{6}{3} = 2 \quad \text{(not between 2.645 and 2.828)}
\]

From the calculations, the only option that lies between \( \sqrt{7} \) and \( \sqrt{8} \) is \( \frac{19}{7} \).

Thus, the rational number that comes between \( \sqrt{7} \) and \( \sqrt{8} \) is **19/7**.