Question

What is a rational number with a denominator of seven that is between square root seven and square root eight write your answers as an improper fraction

Answer 1

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

Calculating these square roots:

1. \(\sqrt{7} \approx 2.6457513110645906\)
2. \(\sqrt{8} = 2\sqrt{2} \approx 2.8284271247461903\)

Next, we want to find a rational number of the form \(\frac{n}{7}\) that is between these two values.

To find appropriate bounds for \(n\):

\[
2.6457513110645906 < \frac{n}{7} < 2.8284271247461903
\]

Multiplying all parts of the inequality by 7 gives:

\[
7 \cdot 2.6457513110645906 < n < 7 \cdot 2.8284271247461903
\]

Calculating the boundaries:

1. \(7 \cdot 2.6457513110645906 \approx 18.52025817745215\)
2. \(7 \cdot 2.8284271247461903 \approx 19.798993873223333\)

This means:

\[
18.52025817745215 < n < 19.798993873223333
\]

Since \(n\) must be an integer, the only possible integer value of \(n\) that lies in this range is \(19\).

Thus, we can write the rational number as:

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

Therefore, the answer is \(\frac{19}{7}\).