Question

Using rational approximations, what statement is true? (1 point)
71 >
−− √ 61
−− √
81 < 9
−− √
81 > 9
−− √
71 <
−− √ 61
−−

Answer 1

To evaluate the given inequalities, let's approximate the square roots of the numbers involved.

1. **Estimate \(\sqrt{61}\)**:
- Since \(61\) is between \(49\) (\(\sqrt{49} = 7\)) and \(64\) (\(\sqrt{64} = 8\)), we know \(7 < \sqrt{61} < 8\).
- A closer approximation is \(\sqrt{61} \approx 7.81\).

2. **Calculate \(\sqrt{81}\)**:
- \(\sqrt{81} = 9\).

Now we can evaluate each statement:

1. **\(71 > \sqrt{61}\)**:
- Since \(71\) is greater than \(\sqrt{61} \approx 7.81\), this statement is **true**.

2. **\(81 < \sqrt{9}\)**:
- This does not hold since \(\sqrt{81} = 9\).

3. **\(81 > \sqrt{9}\)**:
- \(\sqrt{9} = 3\), and \(81 > 3\) is **true**.

4. **\(71 < \sqrt{61}\)**:
- Since we estimated \(\sqrt{61} \approx 7.81\), this statement is **false**.

Based on these evaluations, the true statements are:
- **71 > \(\sqrt{61}\)**
- **81 > \(\sqrt{9}\)**

However, the only statement that stands out as universally true given the context of the question is the first one:

**\(71 > \sqrt{61}\)** is a true statement.