Question

What is a rational number with a denominator of 7 that is between 7–√
7
and 8–√
8
. Write your answer as an improper fraction.(1 point)

Answer 1

To identify a rational number with a denominator of 7 that lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \), we first need to evaluate \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \).

1. First, compute \( \sqrt{7} \) and \( \sqrt{8} \):
- \( \sqrt{7} \approx 2.64575 \)
- \( \sqrt{8} = 2\sqrt{2} \approx 2.82843 \)

2. Now calculate the bounds:
- \( 7 - \sqrt{7} \approx 7 - 2.64575 \approx 4.35425 \)
- \( 8 - \sqrt{8} \approx 8 - 2.82843 \approx 5.17157 \)

So we're looking for a rational number with a denominator of 7 that is between approximately 4.35425 and 5.17157.

Any rational number with a denominator of 7 can be expressed in the form \( \frac{n}{7} \) where \( n \) is an integer.

Next, we will determine the integer values of \( n \) that satisfy:

\[
4.35425 < \frac{n}{7} < 5.17157
\]

Multiplying through by 7 gives us:

\[
4.35425 \times 7 < n < 5.17157 \times 7
\]

Calculating the bounds:

\[
4.35425 \times 7 \approx 30.48175
\]

\[
5.17157 \times 7 \approx 36.201
\]

So, we set:

\[
30.48175 < n < 36.201
\]

The integer values of \( n \) within this range are 31, 32, 33, 34, 35, and 36.

Choosing one of the integer values, let's take \( n = 34 \).

Thus, the rational number is:

\[
\frac{34}{7}
\]

To ensure that this number lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{8} \):

- Checking \( 7 - \sqrt{7} \):
\[
\frac{34}{7} \approx 4.85714
\]

- Checking \( 8 - \sqrt{8} \):
\[
\frac{34}{7} \approx 4.85714
\]

Confirming both inequalities:
\[
4.35425 < 4.85714 < 5.17157
\]

Thus, the rational number you are looking for is:

\[
\frac{34}{7}
\]