Question

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

71 −−−√<61−−√
start root 71 end root less than start root 61 end root

71−−√>61−−√
start root 71 end root greater than start root 61 end root

81−−√>9
start root 81 end root greater than 9

81−−√<9
start root 81 end root less than 9///

Answer 1

are you sure

Answer 2

To evaluate the statements using rational approximations, let's calculate the approximate values step by step.

1. For \(\sqrt{71}\) and \(\sqrt{61}\):
- \(\sqrt{71} \approx 8.426\)
- \(\sqrt{61} \approx 7.810\)

So, \(\sqrt{71} > \sqrt{61}\), which means the statement \( \sqrt{71} < \sqrt{61} \) is false.

2. For \(\sqrt{81}\) and \(9\):
- \(\sqrt{81} = 9\)

So, \(\sqrt{81} = 9\). Therefore:
- The statement \( \sqrt{81} > 9 \) is false.
- The statement \( \sqrt{81} < 9 \) is also false.

In conclusion, the true statement from the options given is:
- \(\sqrt{71} > \sqrt{61}\)

Since that specific option is not explicitly listed, no formal correct response can be selected from the provided options.

Answer 3

Let's re-evaluate the statements carefully:

1. **Comparing \(\sqrt{71}\) and \(\sqrt{61}\)**:
- We know \(\sqrt{71} \approx 8.426\)
- And \(\sqrt{61} \approx 7.810\)

Since \(8.426 > 7.810\), it follows that:
\[
\sqrt{71} > \sqrt{61}
\]

2. **Comparing \(\sqrt{81}\) and \(9\)**:
- As previously noted, \(\sqrt{81} = 9\).

This gives us:
\[
\sqrt{81} = 9
\]

Since these evaluations show:
- \(\sqrt{71} > \sqrt{61}\)
- \(\sqrt{81} = 9\)

From the given options, the only accurate statement is:
- \(\sqrt{71} > \sqrt{61}\)

However, **the true true option is**:
- \(\sqrt{81} = 9\)

As there isn't an option that says \(\sqrt{81} = 9\), the false comparisons were made.

The only incorrect options you gave were:
- \(\sqrt{81} < 9\) and \(\sqrt{81} > 9\), both of which are false.

So, out of the available options, the first one is true:

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

I apologize for the previous oversight. Thank you for allowing me to clarify.