Question

Using rational approximations, what is an inequality statement comparing 21−−√ and 17−−√? Enter <, >, or =

Answer 1

To compare \( \sqrt{21} \) and \( \sqrt{17} \) using rational approximations, we can start by estimating their square roots.

1. We know that \( \sqrt{16} = 4 \) and \( \sqrt{25} = 5 \). This means:
\[
4 < \sqrt{17} < 5
\]

2. For \( \sqrt{21} \):
\[
\sqrt{21} > 4 \quad \text{(since \( 21 > 16 \))}
\]
\[
\sqrt{21} < 5 \quad \text{(since \( 21 < 25 \))}
\]
Therefore:
\[
4 < \sqrt{21} < 5
\]

Now we can use rational approximations to get a better idea of the values.

If we calculate more precise approximations:
- \( \sqrt{17} \) is approximately \( 4.123 \).
- \( \sqrt{21} \) is approximately \( 4.583 \).

Since \( \sqrt{21} \approx 4.583 \) is greater than \( \sqrt{17} \approx 4.123 \), we can conclude:
\[
\sqrt{21} > \sqrt{17}
\]

Thus, the answer is:
\[
\sqrt{21} > \sqrt{17}
\]