Question

Using rational approximations, what is an inequality statement comparing square root eighty one and square root eighty eight

Answer 1

To compare \(\sqrt{81}\) and \(\sqrt{88}\), we first note their exact values.

Calculating \(\sqrt{81}\):
\[
\sqrt{81} = 9
\]

Now calculating \(\sqrt{88}\):
To find \(\sqrt{88}\), we note that \(88\) is between \(81\) and \(100\) (since \(100 = 10^2\)):
\[
\sqrt{81} < \sqrt{88} < \sqrt{100}
\]
So, we know that:
\[
9 < \sqrt{88} < 10
\]

To approximate \(\sqrt{88}\) more closely, we can use rational approximations. Since \(88 = 4 \times 22\), we can express \(\sqrt{88}\) as:

\[
\sqrt{88} = \sqrt{4 \times 22} = 2\sqrt{22}
\]

Next, we can find an approximate value for \(\sqrt{22}\). A reasonable estimate might be:
\(\sqrt{22} \approx 4.69\) (since \(4.69^2 \approx 22\)).

Thus, we have:
\[
\sqrt{88} \approx 2 \times 4.69 \approx 9.38
\]

This gives us a more precise comparison:
\[
9 < \sqrt{88} \approx 9.38
\]

To also verify through direct comparison:
Calculating \(88\) directly:
\[
\sqrt{88} \approx 9.38
\]

Therefore, we can state the inequality clearly:
\[
\sqrt{81} < \sqrt{88}
\]

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