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: \[ \sqrt{81} < \sqrt{88} \]