To estimate \(\sqrt{14}\), we can start with the given information and fill in the blanks.
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"estimate between two whole numbers" - \(\sqrt{14}\) is between 3 and 4 because \(3^2 = 9\) and \(4^2 = 16\).
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"estimate further to the decimal place" - We can estimate the decimal further. We know it's between 3 and 4.
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For the inequality \(3.7^2 < 14 < ___^2\), we calculate \(3.7^2\): \[ 3.7^2 = 13.69 \] Hence, \(13.69 < 14\) is true. For \(14 < x^2\), we need to find \(x\) such that \(x^2 = 16\), so \(x\) would have to be slightly more than 4 (the nearest perfect square). Thus, we can use: \[ \sqrt{14} < 4 \] So, we can fill in this blank with 4.
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Finally, "√14 is between 3.74 and 3.75". We can estimate \(\sqrt{14}\) to two decimal places as approximately 3.74 or 3.75. The more precise calculation shows: \[ 3.74^2 = 13.9876\quad \text{and} \quad 3.75^2 = 14.0625 \] Hence, \(\sqrt{14}\) is indeed between \(3.74\) and \(3.75\).
Putting it all together, we have:
- estimate between two whole numbers: 3 and 4
- estimate further to the decimal place: 3.74 and 3.75
- \(3.7^2 < 14 < 4^2\)
- \(\sqrt{14}\) is between 3.74 and 3.75.
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