First, let's find the approximate values of the square roots:
- \(\sqrt{6} \approx 2.449\)
- \(\sqrt{8} \approx 2.828\)
We are looking for a rational number and an irrational number that lie between these two values.
Now let's evaluate the options:
A. 2.6 as a repeating decimal and the square root of 7
- \(2.6\) (or \(2.666...\)) is a rational number and it is between \(2.449\) and \(2.828\).
- \(\sqrt{7} \approx 2.645751\), which is also between \(2.449\) and \(2.828\) and is irrational.
B. 9/4, square root of 7
- \(9/4 = 2.25\), which is less than \(2.449\).
- \(\sqrt{7} \approx 2.645751\), which is correct.
C. 2.7, square root of 9
- \(2.7\) is a rational number and is between \(2.449\) and \(2.828\).
- \(\sqrt{9} = 3\), which is not between \(2.449\) and \(2.828\).
D. 2.5, pi
- \(2.5\) is a rational number and is between \(2.449\) and \(2.828\).
- \(\pi \approx 3.14159\), which is greater than \(2.828\).
Among these options, only option A includes both a rational and an irrational number within the range between \(\sqrt{6}\) and \(\sqrt{8}\).
So, the correct answer is A. 2.6 as a repeating decimal and the square root of 7.
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