Using rational approximations, what statement is true?(1 point) Responses 71−−√>61−−√ start root 71 end root greater than start root 61 end root 71 −−−√<61−−√ start root 71 end root less than start root 61 end root 81−−√>9 start root 81 end root greater than 9 81−−√<9

Answer 1

To determine which of the statements about the square roots is true, let's evaluate each comparison using rational approximations.

1. \( \sqrt{71} > \sqrt{61} \):

   • The decimal approximations are:
     • \( \sqrt{71} \approx 8.426 \)
     • \( \sqrt{61} \approx 7.810 \)
   • Since \( 8.426 > 7.810 \), this statement is true.

2. \( \sqrt{71} < \sqrt{61} \):

   • This is the opposite of the first statement. Since we established that \( \sqrt{71} > \sqrt{61} \), this one is false.

3. \( \sqrt{81} > 9 \):

   • We know \( \sqrt{81} = 9 \).
   • Therefore, \( 9 > 9 \) is false.

4. \( \sqrt{81} < 9 \):

   • Again, since \( \sqrt{81} = 9 \), this means \( 9 < 9 \) is false.

Based on this analysis, the true statement is:

\[ \sqrt{71} > \sqrt{61} \]