Which of the following is an irrational number between 5 and 7: 6√, 12√, 27√, 36√?

Answer 1

To determine if these numbers are irrational, we need to determine if their square roots are irrational numbers.

The square root of 6 is irrational, so 6√ contains an irrational number.

The square root of 12 is also irrational, so 12√ contains an irrational number.

The square root of 27 is 3√(3), which is irrational since √(3) is irrational. Thus, 27√ contains an irrational number.

The square root of 36 is a whole number, which is rational. Thus, 36√ is rational.

Therefore, the irrational numbers between 5 and 7 are 6√, 12√, and 27√.

Answer 2

What is the largest integer whose square root is an irrational number between 3 and 4?

Answer 3

To find the largest integer whose square root is an irrational number between 3 and 4, we need to find the largest perfect square number between 9 and 16.

The largest perfect square number between 9 and 16 is 16, which is the square of 4. The square root of 16 is 4, which is a rational number.

Therefore, there is no integer between 3 and 4 whose square root is an irrational number.