To compare \(\sqrt{81}\) and \(\sqrt{88}\) using rational approximations, we first find the exact values of the square roots:
\[ \sqrt{81} = 9 \]
Next, we can estimate \(\sqrt{88}\) by recognizing that it falls between two perfect squares: \(81\) and \(100\). We know:
\[ \sqrt{81} = 9 \quad \text{and} \quad \sqrt{100} = 10 \]
To narrow down our approximation of \(\sqrt{88}\), we can calculate a few rough estimates.
Calculating \(9.3^2\) and \(9.4^2\) allows us to find the range:
\[ 9.3^2 = 86.49 \] \[ 9.4^2 = 88.36 \]
Since \(86.49 < 88 < 88.36\), it follows that:
\[ 9.3 < \sqrt{88} < 9.4 \]
Thus, \(\sqrt{88}\) is slightly greater than \(9.3\) and less than \(9.4\). To summarize, we now have:
\[ 9 < \sqrt{88} < 9.4 \]
Since \(9 < 9.4\), we can conclude that:
\[ \sqrt{81} < \sqrt{88} \]
Therefore, the inequality statement comparing the square roots is:
\[ \sqrt{81} < \sqrt{88} \]
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