To analyze the inequalities given, we first need to determine the approximate values of the square roots involved.
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Calculate \(\sqrt{48}\): \[ \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \] Using a rational approximation for \(\sqrt{3}\) (approximately 1.732): \[ \sqrt{48} \approx 4 \times 1.732 \approx 6.928 \]
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Calculate \(\sqrt{36}\): \[ \sqrt{36} = 6 \]
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Calculate \(\sqrt{49}\): \[ \sqrt{49} = 7 \]
Now, we can evaluate the inequalities:
- \( \sqrt{48} < \sqrt{36} \) translates to approximately \( 6.928 < 6 \), which is false.
- \( \sqrt{48} < \sqrt{49} \) translates to approximately \( 6.928 < 7 \), which is true.
- \( \sqrt{36} < \sqrt{49} \) translates to \( 6 < 7 \), which is also true.
Next, let's summarize the evaluations:
- We have established that \( \sqrt{48} \) is greater than \( \sqrt{36} \) and less than \( \sqrt{49} \).
- This implies that \( \sqrt{48} \) is indeed between \( \sqrt{36} \) and \( \sqrt{49} \).
Therefore, the true statements derived from the approximations are:
- \( \sqrt{36} < \sqrt{48} < \sqrt{49} \)
We can also state:
- \( \sqrt{48} > \sqrt{36} \)
- \( \sqrt{49} > 7 \)
The logical conclusion based on the calculated approximate values is: \[ \sqrt{48} > \sqrt{36} \quad \text{and} \quad \sqrt{48} < \sqrt{49} \]
Thus, the valid statements regarding these square roots would be:
- \(\sqrt{48} > \sqrt{36}\)
And the relevant part of your options suggests:
- \(48^{0.5} > 36^{0.5}\) and \(49^{0.5} > 7\)
So the statement "48−−√>36−−√" is true, and "49−−√>7" is also true.
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