Question

Which of these is a rational number that comes between 7–√
7
and 8–√
8
?(1 point)
Responses

56
5
6
5 sixths

197
19
7
19 sevenths

34
3
4
3 fourths

63
6
3
6 thirds

Answer 1

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

1. **Calculate \( \sqrt{7} \)**:
\[
\sqrt{7} \approx 2.64575 \quad \text{(using a calculator)}
\]
Thus,
\[
7 - \sqrt{7} \approx 7 - 2.64575 \approx 4.35425
\]

2. **Calculate \( \sqrt{8} \)**:
\[
\sqrt{8} = 2\sqrt{2} \approx 2.82843 \quad \text{(using a calculator)}
\]
Thus,
\[
8 - \sqrt{8} \approx 8 - 2.82843 \approx 5.17157
\]

Now we check which of the given rational numbers lies between \(4.35425\) and \(5.17157\):

- \(5\) is between \(4.35425\) and \(5.17157\).
- \(6\) is not.
- \(56\) is not.
- \(5\) sixths is approximately \(0.8333\) which is not between the two.
- \(19\) is not.
- \(7\) is not.
- \(19\) sevenths is approximately \(2.7143\) which is not between the two.
- \(3\) is not.
- \(4\) is not.
- \(3\) fourths is \(0.75\) which is not between the two.
- \(63\) is not.
- \(6\) is not.
- \(3\) is not.
- \(6\) thirds equals \(2\) which is not between the two.

Thus, the only rational number that comes between \(7 - \sqrt{7}\) and \(8 - \sqrt{8}\) is \(\boxed{5}\).