Using rational approximations, what is an inequality statement comparing 81−−√

Question

and 8–√8
? Enter <, >, or =.

Answer 1

To compare \( \sqrt{81} \) and \( 8\sqrt{8} \), we can first compute their values.

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

2. Calculate \( 8\sqrt{8} \): First, we find \( \sqrt{8} \): \[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}. \] Therefore, \[ 8\sqrt{8} = 8 \times 2\sqrt{2} = 16\sqrt{2}. \] Now, we need to find \( \sqrt{2} \) to approximate \( 16\sqrt{2} \). A common estimation for \( \sqrt{2} \) is approximately 1.414. Thus, \[ 16\sqrt{2} \approx 16 \times 1.414 \approx 22.624. \]

Now we can compare \( 9 \) and \( 22.624 \): \[ 9 < 22.624. \]

Thus, the inequality relationship is: \[ \sqrt{81} < 8\sqrt{8}. \]

The final answer is: \[ <. \]