Which of the following options includes a rational number and an irrational number between 6–√

Question

and 8–√
?(1 point)
Responses

2.5, π
2.5, π

2.7
,9–√
2 point 7, square root of 9

94
, 7–√
9 fourths, square root of 7

2.6¯¯¯
, 7–√

Answer 1

To find an option that includes a rational number and an irrational number between $\sqrt{6}$ and $\sqrt{8}$, we first need to determine the approximate values of $\sqrt{6}$ and $\sqrt{8}$:

• $\sqrt{6}\approx 2.45$
• $\sqrt{8}\approx 2.83$

So, we need a rational number and an irrational number that lie between approximately 2.45 and 2.83.

Now, let's analyze the given options:

1. 2.5, π:

   • 2.5 is rational, but π (approximately 3.14) is not between $\sqrt{6}$ and $\sqrt{8}$.
   • This option is not valid.

2. 2.7, $\sqrt{9}$:

   • 2.7 is rational, but $\sqrt{9}=3$ is also not between $\sqrt{6}$ and $\sqrt{8}$.
   • This option is not valid.

3. 94, $\sqrt{7}$:

   • 94 is an integer (rational), but $\sqrt{7}\approx 2.64575131106$ is indeed an irrational number and is between $\sqrt{6}$ and $\sqrt{8}$.
   • However, 94 itself is not between $\sqrt{6}$ and $\sqrt{8}$, so this option is not valid.

4. 2.6̅, $\sqrt{7}$:

   • 2.6̅, or $2.\bar{6}$ (which equals $2.666...$), is rational and lies between $\sqrt{6}$ and $\sqrt{8}$.
   • $\sqrt{7}\approx 2.64575131106$ is irrational and also lies between $\sqrt{6}$ and $\sqrt{8}$.

Therefore, the valid option that includes a rational number and an irrational number between $\sqrt{6}$ and $\sqrt{8}$ is:

2.6̅, $\sqrt{7}$.