Question

Using rational approximations, what statement is true?(1 point) 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 compare the values using rational approximations, let's analyze each statement.

1. \( \sqrt{81} \):
\[
\sqrt{81} = 9
\]
Therefore, the statement \( \sqrt{81} > 9 \) is **false**, and \( \sqrt{81} < 9 \) is also **false**.

2. \( \sqrt{71} \) and \( \sqrt{61} \):
- First, we know that \( 8^2 = 64 \) and \( 7^2 = 49 \), so \( \sqrt{61} \) is between 7 and 8.
- Next, \( 8^2 = 64 \) and \( 9^2 = 81 \), which means \( \sqrt{71} \) is also between 8 and 9.

To get more approximate values:
- \( \sqrt{61} \) is roughly \( 7.81 \) (since \( 7.81^2 \approx 61 \)).
- \( \sqrt{71} \) is roughly \( 8.43 \) (since \( 8.43^2 \approx 71 \)).

Now we can compare:
\[
\sqrt{71} \approx 8.43 > \sqrt{61} \approx 7.81
\]

Therefore, \( \sqrt{71} > \sqrt{61} \) is **true**.

In summary, the true statement is:
\[
\sqrt{71} > \sqrt{61}
\]