To determine which integer \( n \) is close to the square of \( \frac{9}{4} \), we first calculate \( \left(\frac{9}{4}\right)^2 \):
\[ \left(\frac{9}{4}\right)^2 = \frac{9^2}{4^2} = \frac{81}{16} \]
Next, we can convert \( \frac{81}{16} \) into a decimal:
\[ \frac{81}{16} = 5.0625 \]
Now, we look for the closest integer to \( 5.0625 \). The integers closest to \( 5.0625 \) are \( 5 \) and \( 6 \).
To confirm, we check the square roots:
- The integer \( 5 \) has a square root of \( \sqrt{5} \approx 2.236 \)
- The integer \( 6 \) has a square root of \( \sqrt{6} \approx 2.449 \)
Next, we find which integer (5 or 6) is closer to \( \frac{9}{4} \approx 2.25 \):
- The distance from \( 2.25 \) to \( \sqrt{5} \approx 2.236 \) is \( |2.25 - 2.236| = 0.014 \)
- The distance from \( 2.25 \) to \( \sqrt{6} \approx 2.449 \) is \( |2.25 - 2.449| = 0.199 \)
Thus, \( \frac{9}{4} \approx 2.25 \) is closer to \( \sqrt{5} \).
Conclusion: 9/4 is close to the square root of the integer 5.
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