To compare \( \sqrt{21} \) and \( \sqrt{17} \) using rational approximations, we can calculate the approximate values of each square root.
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The square root of 21 can be approximated as follows: \[ \sqrt{21} \approx 4.58 \quad (\text{since } 4.58^2 = 20.9764 \text{ and } 4.59^2 = 20.0881) \]
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The square root of 17 can be approximated similarly: \[ \sqrt{17} \approx 4.12 \quad (\text{since } 4.12^2 = 16.9744 \text{ and } 4.13^2 = 17.0569) \]
Now, comparing these two values: \[ 4.58 > 4.12 \]
Therefore, we can conclude: \[ \sqrt{21} > \sqrt{17} \]
The inequality is: \[ \sqrt{21} > \sqrt{17} \]
So the final statement is: \[ \sqrt{21} > \sqrt{17} \]