To find the value of \(\sqrt{68}\) and identify the interval it falls into, we first compute the square root:
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Calculate \(\sqrt{68}\): \[ \sqrt{68} = \sqrt{(4 \times 17)} = \sqrt{4} \times \sqrt{17} = 2\sqrt{17} \]
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Next, we need to estimate \(\sqrt{17}\). We know that: \[ 4^2 = 16 < 17 < 25 = 5^2 \] Therefore, \(4 < \sqrt{17} < 5\).
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To find a better approximation for \(\sqrt{17}\), we will calculate the squares of numbers between 4 and 5:
- \(4.1^2 = 16.81\)
- \(4.2^2 = 17.64\)
So, \(4.1 < \sqrt{17} < 4.2\).
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By using this approximation: \[ 2 \times 4.1 < 2\sqrt{17} < 2 \times 4.2 \] This gives us: \[ 8.2 < \sqrt{68} < 8.4 \]
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Thus, \(\sqrt{68}\) is approximately between \(8.2\) and \(8.4\).
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Now, rounding \(\sqrt{68}\) to the nearest half:
- The nearest half values in this range are 8.0, 8.5 (because \(8.2\) is closer to \(8.0\) than \(8.5\)).
Therefore, \(\sqrt{68}\) is between \(8.5\) and \(9\).
The final answer is: between 8.5 and 9.