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 simplified

Answer 1

To find a rational number with a denominator of 7 that lies between \(\sqrt{7}\) and \(\sqrt{8}\), we first calculate the approximate values of \(\sqrt{7}\) and \(\sqrt{8}\):

- \(\sqrt{7} \approx 2.645751311\)
- \(\sqrt{8} \approx 2.828427124\)

Next, we look for a rational number in the form of \(\frac{n}{7}\) where \(n\) is an integer that lies between these two values.

To find suitable \(n\), we need to solve the inequalities:

\[
2.645751311 < \frac{n}{7} < 2.828427124
\]

Multiplying the entire inequality by 7 to eliminate the denominator results in:

\[
7 \cdot 2.645751311 < n < 7 \cdot 2.828427124
\]

Calculating the products gives:

\[
18.520258177 < n < 19.798989868
\]

The only integer \(n\) that lies within this range is \(n = 19\).

Thus, the rational number we seek is:

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

This fraction is already in its simplest form and is between \(\sqrt{7}\) and \(\sqrt{8}\).

So the final answer is:

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