Which of the following options includes a rational number and an irrational number between 6–√ and 8–√ ?(1 point) Responses 94 , 7–√ 9 fourths, square root of 7 2.5, π 2.5, π 2.7 ,9–√ 2 point 7, square root of 9 2.6¯¯¯ , 7–√ 2 point repeating 6, square root of 7

1 answer

To find a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \), we first need to approximate these square roots:

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

Now, we can analyze the provided options:

  1. 9, \( \sqrt{7} \): \( \sqrt{7} \) is irrational (approximately \( 2.645751 \)), while \( 9 \) is rational.
  2. 9/4, \(\sqrt{7}\): \( 9/4 = 2.25 \) is rational and is less than \( \sqrt{6} \). Thus, this pair is not valid.
  3. 2.5, \( \pi \): \( 2.5 \) is rational (between \( \sqrt{6} \) and \( \sqrt{8} \)), and \( \pi \) is irrational (approximately \( 3.14 \)), so this is not within our range.
  4. 2.7, \( \sqrt{9} \): \( 2.7 \) is rational, and \( \sqrt{9} = 3 \), which is greater than \( \sqrt{8} \). Thus, this pair is not valid.
  5. 2.6, \( 7 \sqrt{} \): \( 2.6 \) is rational (between \( \sqrt{6} \) and \( \sqrt{8} \)), yet \( 7 \sqrt{} \) is not a valid number.
  6. 2.6̅, \( \sqrt{7} \): \( 2.6̅ \) (or \( 2.666... \)) is rational and \( \sqrt{7} \) is irrational. While this option includes an irrational number, the rational number must be evaluated in context (likely between the bounds). Since \( \sqrt{7} \) is about \( 2.645751 \), it fits.
  7. 2.5, \( \sqrt{7} \): As stated before \( \sqrt{7} \) is valid.

Options of combinations must be evaluated.

Would \( 2.6̅, \sqrt{7} \) completely validate as the only selection staying within the threshold of \( \sqrt{6} \) and \( \sqrt{8} \)? In ending, the effective choice here would be through 2.5, \(\sqrt{7}\) working together fitting within the targets.

In conclusion, the viable response is composed by pairing items \( 2.6̅ \) and \( \sqrt{7} \)—accompanied within bounds of rationality and the irrationally catered experience. Or comprehensively, opt towards variants within options merely offering clear simplicities attributions.