To find the square root of 58, we note that it lies between two perfect squares:
- \( 7^2 = 49 \)
- \( 8^2 = 64 \)
Therefore,
\[ 7 < \sqrt{58} < 8 \]
Next, we can determine the approximate value of \( \sqrt{58} \). Since \( 58 \) is closer to \( 64 \) than to \( 49 \), we can estimate that \( \sqrt{58} \) is closer to \( 8 \).
Calculating the square roots of numbers around 58 gives:
- \( 7.5^2 = 56.25 \)
- \( 7.6^2 = 57.76 \)
- \( 7.7^2 = 59.29 \)
Since \( 58 \) is between \( 7.6^2 \) and \( 7.7^2 \), we can conclude that:
\[ 7.6 < \sqrt{58} < 7.7 \]
Thus, when rounding \( \sqrt{58} \) to the nearest half:
- \( 7.5 \) rounds up to \( 8 \),
- \( 7.6 \) rounds down to \( 7.5 \),
- \( 7.7 \) rounds down to \( 7.5 \).
Therefore, the value of \( \sqrt{58} \) when rounded to the nearest half is approximately \( 7.5 \).
To summarize:
- The whole number \( \sqrt{58} \) is between is \( 7 \) and \( 8 \).
- The rational number rounded to the nearest half is \( 7.5 \).
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