Which of the following options includes a rational number and an irrational number between 6–√ and 8–√?
1) 2.5, π
2) 2.6¯¯¯, 7–√
3) 2.7, 9–√
4) 94, 7–√
The number 6–√ is approximately 4.77.
Option 2) includes the rational number 2.6¯¯¯ which is less than 4.77 and the irrational number 7–√ which is greater than 4.77.
So, the correct option is 2) 2.6¯¯¯, 7–√.
To determine which option includes a rational number and an irrational number between 6–√ and 8–√, we need to simplify the given expressions.
Let's calculate the values for each option:
1) 2.5, π
The value of 6–√ is approximately 3.54, and the value of 8–√ is approximately 4.59.
Since both 2.5 and π are rational numbers, option 1 does not include both a rational and an irrational number between 6–√ and 8–√.
2) 2.6¯¯¯, 7–√
The value of 6–√ is approximately 3.54, and the value of 8–√ is approximately 4.59.
Since 2.6¯¯¯ (which means 2.6666...) is rational and 7–√ is irrational, option 2 includes both a rational and an irrational number between 6–√ and 8–√.
3) 2.7, 9–√
The value of 6–√ is approximately 3.54, and the value of 8–√ is approximately 4.59.
Since both 2.7 and 9–√ are rational numbers, option 3 does not include both a rational and an irrational number between 6–√ and 8–√.
4) 94, 7–√
The value of 6–√ is approximately 3.54, and the value of 8–√ is approximately 4.59.
Since 94 is a rational number and 7–√ is irrational, option 4 includes both a rational and an irrational number between 6–√ and 8–√.
Therefore, the correct answer is option 2) 2.6¯¯¯, 7–√.