To find a pair that includes both a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \), we first need to approximate the square roots:
- \( \sqrt{6} \) is approximately \( 2.45 \).
- \( \sqrt{8} \) is approximately \( 2.83 \).
Thus, we are looking for a rational and an irrational number in the range \( (2.45, 2.83) \).
Let's evaluate the options:
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9, \( \sqrt{7} \) - \( \sqrt{7} \) is irrational and approximately \( 2.6457 \), which is between \( \sqrt{6} \) and \( \sqrt{8} \). 9 is rational but not in the required range.
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\( 9/4 \), \( \sqrt{7} \) - \( 9/4 = 2.25 \) is rational but not in the range. \( \sqrt{7} \) as noted is irrational.
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2.7, \( \sqrt{9} \) - 2.7 is rational and \( \sqrt{9} = 3 \) is also rational, so this doesn't work.
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2.5, \( \pi \) - 2.5 is rational and \( \pi \) is irrational, so this pair works.
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2.6, \( \sqrt{7} \) - 2.6 is rational, and \( \sqrt{7} \) is irrational.
From the evaluation, the pairs that contain a rational number and an irrational number in the specified range are:
- 2.5, \( \pi \)
- 2.6, \( \sqrt{7} \)
So, the correct answer is:
2.5, \( \pi \) and 2.6, \( \sqrt{7} \) both contain a rational and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \).
3 answers