Bot GPT 4-o mini To compare \(\sqrt{42}\) and \(\sqrt{58}\) and find the difference between their approximate values, we can first estimate each square root.
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Estimate \(\sqrt{42}\):
- We know that \(6^2 = 36\) and \(7^2 = 49\), so \(6 < \sqrt{42} < 7\).
- To get a closer estimate, we can use approximate values: \[ \sqrt{42} \approx 6.4\ \text{(since } 6.4^2 = 40.96 \text{, which is close to 42)} \]
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Estimate \(\sqrt{58}\):
- We know that \(7^2 = 49\) and \(8^2 = 64\), so \(7 < \sqrt{58} < 8\).
- For a closer approximation: \[ \sqrt{58} \approx 7.6\ \text{(since } 7.6^2 = 57.76 \text{, which is close to 58)} \]
Now that we have estimated both square roots:
- \(\sqrt{42} \approx 6.4\)
- \(\sqrt{58} \approx 7.6\)
Next, we can calculate the difference: \[ \sqrt{58} - \sqrt{42} \approx 7.6 - 6.4 = 1.2 \]
To find the approximate difference in tenths:
- Since \(1.2\) can also be written as \(12\) tenths, the approximate difference in tenths between \(\sqrt{58}\) and \(\sqrt{42}\) is 12 tenths.
Thus, the approximate difference in tenths between \(\sqrt{42}\) and \(\sqrt{58}\) is \(\boxed{1.2}\).
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