Using rational approximations, what statement is true?(1 point)

Question

Responses

81−−√>9
start root 81 end root greater than 9
81−−√<9
start root 81 end root less than 9
71−−√>61−−√
start root 71 end root greater than start root 61 end root
71 −−−√<61−−√

Answer 1

To determine the truth of each statement using rational approximations, let's evaluate them one at a time.

1. \(\sqrt{81} > 9\):
   \(\sqrt{81} = 9\). Therefore, this statement is false.

2. \(\sqrt{81} < 9\):
   As established, \(\sqrt{81} = 9\). Therefore, this statement is also false.

3. \(\sqrt{71} > \sqrt{61}\):
   To evaluate this, we can look at the approximate values:
   \(\sqrt{71} \approx 8.426\) and \(\sqrt{61} \approx 7.810\).
   Since \(8.426 > 7.810\), this statement is true.

4. \(\sqrt{71} < \sqrt{61}\):
   Since we have already established that \(\sqrt{71} > \sqrt{61}\), this statement is false.

Thus, the true statement is \(\sqrt{71} > \sqrt{61}\).