To compare \( \sqrt{21} \) and \( \sqrt{17} \) using rational approximations, we can estimate their square roots.
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Estimate \( \sqrt{21} \):
- \( \sqrt{21} \) is between \( \sqrt{16} = 4 \) and \( \sqrt{25} = 5 \).
- A better approximation can be found by using a calculator or by considering \( 4.5^2 = 20.25 \) and \( 4.6^2 = 21.16 \), so \( 4.58^2 \) can be tested and gives us \( 20.9764 \) which is less than \( 21 \) but \( 4.59^2 = 20.0881 \) which is still less than \( 21 \). Thus, we find that \( \sqrt{21} \approx 4.582 \).
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Estimate \( \sqrt{17} \):
- Similarly, \( \sqrt{17} \) is between \( \sqrt{16} = 4 \) and \( \sqrt{25} = 5 \).
- Using similar calculations, \( 4.1^2 = 16.81 \) and \( 4.2^2 = 17.64 \), hence we can realize that \( 4.12^2 = 16.9744 \) which is less than \( 17 \) and \( 4.13^2 = 17.0569 \) which is more than \( 17 \). Thus, \( \sqrt{17} \approx 4.123 \).
Now, let's compare:
- \( \sqrt{21} \approx 4.582 \)
- \( \sqrt{17} \approx 4.123 \)
Since \( 4.582 > 4.123 \), we can conclude:
\[ \sqrt{21} > \sqrt{17} \]
Thus, the inequality statement is:
\[ \sqrt{21} > \sqrt{17} \]