Using rational approximations, what statement is true?(1 point)%0D%0AResponses%0D%0A%0D%0A81−−√>9%0D%0Astart root 81 end root greater than 9%0D%0A%0D%0A71−−√>61−−√%0D%0Astart root 71 end root greater than start root 61 end root%0D%0A%0D%0A71 −−−√<61−−√%0D%0Astart root 71 end root less than start root 61 end root%0D%0A%0D%0A81−−√<9

Answer 1

To determine which statement is true using rational approximations, let's evaluate each expression:

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

   • \( \sqrt{81} = 9 \)
   • This is false: \( \sqrt{81} = 9 \), so it is not greater than 9.

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

   • The approximate values are \( \sqrt{71} \approx 8.426 \) and \( \sqrt{61} \approx 7.810 \).
   • This is true: \( 8.426 > 7.810 \).

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

   • This contradicts the previous statement where we found \( \sqrt{71} > \sqrt{61} \).
   • This is false.

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

   • As stated earlier, \( \sqrt{81} = 9 \).
   • This is also false: \( \sqrt{81} = 9 \), so it is not less than 9.

Based on the evaluations above, the only true statement is:

\( \sqrt{71} > \sqrt{61} \)