To determine which option includes a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \), we first need to find the approximate values of \( \sqrt{6} \) and \( \sqrt{8} \):
- \( \sqrt{6} \approx 2.45 \)
- \( \sqrt{8} \approx 2.83 \)
Now, we can analyze the provided options:
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\( 2.6\overline{6}, \sqrt{7} \)
- \( 2.6\overline{6} \) is a rational number.
- \( \sqrt{7} \) is an irrational number.
- \( 2.6\overline{6} \approx 2.666... \) and \( \sqrt{7} \approx 2.64575\).
- Both are between \( \sqrt{6} \) and \( \sqrt{8} \).
-
\( 2.5, \pi \)
- \( 2.5 \) is a rational number.
- \( \pi \) is an irrational number.
- \( 2.5 \) is between \( \sqrt{6} \approx 2.45 \) and \( \sqrt{8} \approx 2.83 \), but \( \pi \approx 3.14 \) is not.
-
\( \frac{9}{4}, \sqrt{7} \)
- \( \frac{9}{4} = 2.25 \) (rational number), not between \( \sqrt{6} \) and \( \sqrt{8} \).
- \( \sqrt{7} \) is irrational but doesn’t meet the condition.
-
\( 2.7, \sqrt{9} \)
- \( 2.7 \) is a rational number.
- \( \sqrt{9} = 3 \) is also a rational number.
- \( \sqrt{9} \) is not between \( \sqrt{6} \) and \( \sqrt{8} \).
The only option that includes both a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \) is option 1: \( 2.6\overline{6}, \sqrt{7} \).