Question

what is a rational number with a denominator of 7 that is between radical 7 and radical 8, write your answer as a improper fraction

Answer 1

To find a rational number with a denominator of 7 that is between \(\sqrt{7}\) and \(\sqrt{8}\), we first need to calculate the approximate decimal values of \(\sqrt{7}\) and \(\sqrt{8}\).

1. **Calculate \(\sqrt{7}\) and \(\sqrt{8}\)**:
- \(\sqrt{7} \approx 2.64575\)
- \(\sqrt{8} \approx 2.82843\)

Now, we need to find a rational number in the form \(\frac{n}{7}\) that falls between these two values. This can be done by setting up the inequalities:

\[
\sqrt{7} < \frac{n}{7} < \sqrt{8}
\]

Multiplying all sides of the inequalities by 7 gives us:

\[
7\sqrt{7} < n < 7\sqrt{8}
\]

Calculating \(7\sqrt{7}\) and \(7\sqrt{8}\):

- \(7\sqrt{7} \approx 7 \times 2.64575 \approx 18.52025\)
- \(7\sqrt{8} \approx 7 \times 2.82843 \approx 19.79901\)

This means we need to find integers \(n\) such that:

\[
18.52025 < n < 19.79901
\]

The only integer \(n\) that satisfies this inequality is \(n=19\).

Thus, the rational number with a denominator of 7 that is between \(\sqrt{7}\) and \(\sqrt{8}\) is:

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

This is the improper fraction you were looking for.