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
-6/3

Answer 1

To determine which of the given options is a rational number that falls between the square root of 7 and the square root of 8, we first need to approximate the values of the square roots.

1. \(\sqrt{7} \approx 2.6457\)
2. \(\sqrt{8} \approx 2.8284\)

So, we are looking for a rational number \(x\) such that:

\[
2.6457 < x < 2.8284
\]

Now let's evaluate each option:

1. \( \frac{5}{6} \approx 0.8333\) (does not fall between the two square roots)
2. \( \frac{19}{7} \approx 2.7143\) (falls between the two square roots)
3. \( \frac{3}{4} = 0.75\) (does not fall between the two square roots)
4. \( \frac{6}{3} = 2\) (does not fall between the two square roots)

From these calculations, the only rational number that lies between \(\sqrt{7}\) and \(\sqrt{8}\) is:

\[
\frac{19}{7}
\]