To compare \(\sqrt{7}\) and \(\sqrt{12}\), we can approximate their values.
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Calculate \(\sqrt{7}\):
- We know that \(2.5^2 = 6.25\) and \(2.6^2 = 6.76\) and \(2.7^2 = 7.29\).
- Therefore, \(\sqrt{7} \approx 2.645\).
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Calculate \(\sqrt{12}\):
- We know that \(3.4^2 = 11.56\) and \(3.5^2 = 12.25\).
- Therefore, \(\sqrt{12} \approx 3.464\).
Now we can compare these two values:
- \(\sqrt{7} \approx 2.645\)
- \(\sqrt{12} \approx 3.464\)
Next, we find the approximate difference between \(\sqrt{12}\) and \(\sqrt{7}\):
\[ \sqrt{12} - \sqrt{7} \approx 3.464 - 2.645 = 0.819 \]
Rounding \(0.819\) to the nearest tenth gives:
- The difference is approximately \(0.8\).
Therefore, on the number line, \(\sqrt{12}\) is greater than \(\sqrt{7}\) and the approximate difference in tenths between the two values is 0.8.