Question

Using rational approximations what is an inequality statement comparing the square root of 81and the square root of 88?

Answer 1

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:

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