Question

Which of these is a rational number that comes between the square root of 7 and the square root of 8?
-5/6
-19/7
-3/4
-6/2

Answer 1

To determine which of the given options is a rational number that comes between \(\sqrt{7}\) and \(\sqrt{8}\), we first need to find approximate values for these square roots:

\[
\sqrt{7} \approx 2.6457513110645906 \quad \text{and} \quad \sqrt{8} \approx 2.8284271247461903.
\]

So we are looking for a rational number between \(2.6457513110645906\) and \(2.8284271247461903\).

Now we will evaluate each option:

1. **\(-\frac{5}{6}\)**:
\[
-\frac{5}{6} \approx -0.8333
\]
This number is not between \(\sqrt{7}\) and \(\sqrt{8}\).

2. **\(-\frac{19}{7}\)**:
\[
-\frac{19}{7} \approx -2.7143
\]
This number is also not between \(\sqrt{7}\) and \(\sqrt{8}\).

3. **\(-\frac{3}{4}\)**:
\[
-\frac{3}{4} \approx -0.75
\]
This number is not between \(\sqrt{7}\) and \(\sqrt{8}\).

4. **\(-\frac{6}{2}\)** (which simplifies to \(-3\)):
\[
-3
\]
This number is not between \(\sqrt{7}\) and \(\sqrt{8}\).

Since none of the options are valid rational numbers that fall between \(\sqrt{7}\) and \(\sqrt{8}\), it appears that none of the provided choices satisfy the condition. Therefore, there is no rational number from the given options that lies between \(\sqrt{7}\) and \(\sqrt{8}\). If you have other numbers or options, feel free to provide those.