To find a rational number and an irrational number between \(\sqrt{6}\) and \(\sqrt{8}\):
- Calculate \(\sqrt{6}\) and \(\sqrt{8}\):
- \(\sqrt{6} \approx 2.449\)
- \(\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \approx 2.828\)
Thus, the interval we are considering is approximately \(2.449 < x < 2.828\).
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Identify a rational number in this interval:
- A simple rational number could be \(2.5\) (or \(\frac{5}{2}\)), since it lies between \(2.449\) and \(2.828\).
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Identify an irrational number in this interval:
- An irrational number could be \(\sqrt{7}\), which is approximately \(2.646\), and lies between \(\sqrt{6}\) and \(\sqrt{8}\).
Therefore, the option including a rational number and an irrational number between \(\sqrt{6}\) and \(\sqrt{8}\) could be \(2.5\) (rational) and \(\sqrt{7}\) (irrational).
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