Question

Using rational approximations, what is an inequality statement comparing √81 {answer here} √88 ?

Answer 1

To compare \( \sqrt{81} \) and \( \sqrt{88} \) using rational approximations, we first calculate \( \sqrt{81} \) and \( \sqrt{88} \).

1. Calculate \( \sqrt{81} \):
\[
\sqrt{81} = 9
\]

2. For \( \sqrt{88} \), we can find an approximate decimal value. We know:
\[
88 = 4 \times 22
\]
So, we can express it as:
\[
\sqrt{88} = \sqrt{4 \times 22} = 2\sqrt{22}
\]
Now we'll estimate \( \sqrt{22} \). We know that \( 4^2 = 16 \) and \( 5^2 = 25 \), thus \( \sqrt{22} \) is between 4 and 5.

To get a more accurate approximation, we can further refine it. We can try \( 4.7 \):
\[
4.7^2 = 22.09 \quad (\text{too high})
\]
Now try \( 4.6 \):
\[
4.6^2 = 21.16 \quad (\text{too low})
\]
This means \( \sqrt{22} \) is between \( 4.6 \) and \( 4.7 \). A better approximation is around \( 4.69 \) (since \( 4.69^2 \approx 22 \)).

Therefore,
\[
\sqrt{88} \approx 2 \times 4.69 \approx 9.38
\]

3. Now we can compare \( \sqrt{81} \) and \( \sqrt{88} \):
\[
\sqrt{81} = 9 \quad \text{and} \quad \sqrt{88} \approx 9.38
\]

Thus, the inequality statement is:
\[
\sqrt{81} < \sqrt{88}
\]

In conclusion, the correct inequality is:
\[
9 < \sqrt{88}
\] or simply \( \sqrt{81} < \sqrt{88} \).