Which of the following options includes a rational number and an irrational number between √(6) and √(8)?

Question

A. (9)/(4), √(7)
B. 2.7, √(9)
C. \overline{6}, √(7)
D. 2.5, \pi

Answer 1

To determine which option includes a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \):

1. Calculate \( \sqrt{6} \) and \( \sqrt{8} \):

   • \( \sqrt{6} \) is approximately \( 2.45 \).
   • \( \sqrt{8} \) is approximately \( 2.83 \).

   Therefore, we are looking for one rational number and one irrational number that lie within the interval \( (2.45, 2.83) \).

2. Evaluate the options:

   • Option A: \( \frac{9}{4} \) and \( \sqrt{7} \)

     • \( \frac{9}{4} = 2.25 \) (rational, outside the interval)
     • \( \sqrt{7} \approx 2.645 \) (irrational, inside the interval)

   • Option B: \( 2.7 \) and \( \sqrt{9} \)

     • \( 2.7 \) (rational, inside the interval)
     • \( \sqrt{9} = 3 \) (rational, outside the interval)

   • Option C: \( \overline{6} = 6.\overline{6} \) (6.666...) and \( \sqrt{7} \)

     • \( 6.\overline{6} \) (rational, outside the interval)
     • \( \sqrt{7} \approx 2.645 \) (irrational, inside the interval)

   • Option D: \( 2.5 \) and \( \pi \)

     • \( 2.5 \) (rational, inside the interval)
     • \( \pi \approx 3.14159 \) (irrational, outside the interval)

3. Conclusion: Only Option A \( \frac{9}{4} , and , \sqrt{7} \) contains one rational number (\(\sqrt{7}\)) and one irrational number. However, \(\frac{9}{4}\) is outside the specified range. All other options mix rational and irrational numbers but also fall outside the range.

Thus, the best choice is A, where only one number is irrational within the range.

Answer 2

wrong

Answer 3

Let's reevaluate the options carefully.

We want to find a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \). As established:

• \( \sqrt{6} \approx 2.45 \)
• \( \sqrt{8} \approx 2.83 \)

Now, we will assess each option again:

Option A: \( \frac{9}{4}, \sqrt{7} \)

• \( \frac{9}{4} = 2.25 \) (rational, outside the interval)
• \( \sqrt{7} \approx 2.645 \) (irrational, inside the interval)

Option B: \( 2.7, \sqrt{9} \)

• \( 2.7 \) (rational, inside the interval)
• \( \sqrt{9} = 3 \) (rational, outside the interval)

Option C: \( 6.\overline{6}, \sqrt{7} \)

• \( 6.\overline{6} \) (rational, outside the interval)
• \( \sqrt{7} \approx 2.645 \) (irrational, inside the interval)

Option D: \( 2.5, \pi \)

• \( 2.5 \) (rational, inside the interval)
• \( \pi \approx 3.14159 \) (irrational, outside the interval)

Summary of Findings:

• Option A: One number is irrational, but the rational number is not in the interval.
• Option B: One number is in the interval, but both numbers cannot meet the criteria since \( \sqrt{9} \) is rational and outside the interval.
• Option C: The rational number is outside the interval, while the irrational number is inside.
• Option D: Here, \( 2.5 \) is in the interval and \( \pi \) is outside.

Correct Conclusion:

The only option where the irrational number is indeed in the interval \( ( \sqrt{6}, \sqrt{8} ) \) is Option C which contains \( \sqrt{7} \) as the irrational number.

However, it seems there was an oversight in keeping track of which option provided both a rational and an irrational number correctly fitting the interval criteria since:

• Options with both numbers fit the criteria of one inside:

• We can state that Option A and Option B do not have a corresponding rational number that stays within ( ( \sqrt{6}, \sqrt{8} )**.

So, the best correct pairs that do qualify for being between \(\sqrt{6}\) and \(\sqrt{8}\) are actually Option D: \(2.5, \pi\) as rational and irrational, respectively, given that both recognize \(2.5\) inside the range and \( \pi \) outside.

Thank you for your patience while correcting this!