Question

What is a rational number with a denominator of $ that is between square root of 7 and 8

Answer 1

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

Calculating \(\sqrt{7}\):

\[
\sqrt{7} \approx 2.6457513110645906
\]

We are looking for a rational number \( \frac{p}{d} \) where \(d\) is the denominator, that satisfies:

\[
\sqrt{7} < \frac{p}{d} < 8
\]

Next, we need to establish the bounds. Since \(8\) is an easy whole number, we know that any rational number with a denominator that is a positive integer (let's say 1 for simplicity) will be strictly more than \(\sqrt{7}\) if its numerator \(p\) is varied enough.

Concentrating on rational numbers, consider using a denominator \(d\) of your choice, for example, let’s set \(d = 1\); hence \(d\) must remain a positive integer.

### For \(d = 1\):
Let's check simple integers. For \(p = 3\):
\[
\frac{p}{d} = 3 < 8 \quad \text{and} \quad 3 > \sqrt{7} \quad \text{(Evaluating 2.645751...)}
\]
So here, \(3\) is a rational number that satisfies \(\sqrt{7} < 3 < 8\).

### For \(d = 2\):
Another rational number could be:
\[
\frac{p}{d} = \frac{5}{2} \approx 2.5
\]
However, we realize that: \(2.5\) is not greater than \(\sqrt{7}\) which means that \( \frac{p}{d} \) isn’t valid.

### For \(d = 5\)
Now, choosing \(d = 5\), we can try valid numerators that keep us within the bounds.
Let’s try \(p = 13\):
\[
\frac{13}{5} = 2.6 < 3 \quad (which is okay)
\]
Then \(13\) is usable because \(\frac{13}{5} \approx 2.6\) is validly greater than \(\sqrt{7}\).

### Here’s the general approach:
Let’s summarize the use of \(d = 1\) and \(d = 5\).

Thus, one rational number that has a denominator and satisfies being between \(\sqrt{7}\) and \(8\) is:
\[
\frac{13}{5} \approx 2.6
\]

However, let's also check integers from the greater scope \(d = 100\). By simply decimal expansion:
Let \(d = 100\):
Choose \(p = 265\):
Then \(\frac{265}{100} = 2.65\) which also lies validly.

Therefore the answer would eventually be based on choices:
\[
\frac{13}{5} \text{ or } \frac{265}{100} \text{ are both rational and valid between } \sqrt{7} \text{ and } 8.
\]