The √40 is between _____ and _____.

Answer 1

To find the square root of 40, we can find two perfect squares between which 40 lies.

The perfect squares just below and above 40 are:

• \(6^2 = 36\)
• \(7^2 = 49\)

Since \(36 < 40 < 49\), we can conclude that:

\(\sqrt{40}\) is between \(6\) and \(7\).

Thus, the answer is: 6 and 7.

Answer 2

Using rational approximations, what is the correct inequality statement comparing √81 and √88? Enter <, >, or =.

Answer 3

To compare \(\sqrt{81}\) and \(\sqrt{88}\):

1. Calculate \(\sqrt{81}\): \[ \sqrt{81} = 9 \]

2. Estimate \(\sqrt{88}\):

   • We know that \(88\) is between \(81\) and \(100\) (with \(9^2 = 81\) and \(10^2 = 100\)).
   • Therefore, \(\sqrt{88}\) is between \(9\) and \(10\).
   • To refine our estimate, notice \(88\) is closer to \(81\) than to \(100\).

3. We can compute \(\sqrt{88}\) more accurately: \[ \sqrt{88} \approx 9.38 \quad (\text{since } 9.38^2 \approx 88) \]

Comparing the two values:

• \(\sqrt{81} = 9\)
• \(\sqrt{88} \approx 9.38\)

Thus, we have: \[ \sqrt{81} < \sqrt{88} \]

The correct inequality statement is: \[ \sqrt{81} < \sqrt{88} \]