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

1 answer

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.