Using rational approximations, what is an inequality statement comparing √81 <,> √88 ?

Answer 1

To compare \(\sqrt{81}\) and \(\sqrt{88}\), we can calculate their values or use known approximations.

First, let's evaluate \(\sqrt{81}\): \[ \sqrt{81} = 9 \]

Next, we can approximate \(\sqrt{88}\). We know: \[ \sqrt{88} = \sqrt{4 \times 22} = 2\sqrt{22} \] Now let's approximate \(\sqrt{22}\). We know the perfect squares near 22 are 4 (\(2^2\)) and 25 (\(5^2\)), so \(\sqrt{22}\) is between 4 and 5.

A more precise approximation can be done:

• \(\sqrt{22} \approx 4.69\) (since \(4.69^2 \approx 22\))

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

Now we compare: \[ \sqrt{81} = 9 < 9.38 \approx \sqrt{88} \]

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