To find the least integer whose square root is an irrational number between 5 and 7, we first identify the range of numbers we want to consider:
- The square root of a number \( n \) is irrational between the values \( 5 \) and \( 7 \).
- Squaring the bounds gives us:
- Lower limit: \( 5^2 = 25 \)
- Upper limit: \( 7^2 = 49 \)
Thus, we are looking for integers \( n \) in the range \( 25 < n < 49 \).
Next, we need \( n \) to be an integer that is not a perfect square because the square root must be irrational. The perfect squares between 25 and 49 are:
- \( 25 \) (since \( 5^2 = 25 \))
- \( 36 \) (since \( 6^2 = 36 \))
- \( 49 \) (since \( 7^2 = 49 \))
Now, we identify the integers in the interval \( (25, 49) \) that are not perfect squares. The integers between 25 and 49 are: 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, and 48.
Next, we can see which of these integers is the least integer that satisfies the requirement:
- 26 is not a perfect square.
To confirm, we take the square root of 26: \[ \sqrt{26} \approx 5.099 \] This value is indeed an irrational number between 5 and 7.
Thus, the answer is: \[ \boxed{26} \]
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