Which of the following is an irrational number between 5 and 7: 6√, 12√, 27√, 36√?
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√.
What is the largest integer whose square root is an irrational number between 3 and 4?
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.
To determine which of the given numbers is an irrational number between 5 and 7, we need to find their decimal approximations and check if they are non-repeating and non-terminating.
Let's calculate the decimal approximations for each number:
1. For 6√:
√6 ≈ 2.449
Therefore, 6√ ≈ 6 × 2.449 ≈ 14.694
2. For 12√:
√12 ≈ 3.464
Therefore, 12√ ≈ 12 × 3.464 ≈ 41.57
3. For 27√:
√27 ≈ 5.196
Therefore, 27√ ≈ 27 × 5.196 ≈ 140.212
4. For 36√:
√36 = 6
Therefore, 36√ = 36 × 6 = 216
From the calculations, we can see that all the numbers are rational except for 6√ (approximately 14.694). Therefore, the only irrational number between 5 and 7 from the given options is 6√.