To find the least integer whose square root is an irrational number between 5 and 7, we need to identify the range of numbers whose square roots fit within that criteria.
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The square root of the number must be greater than 5: \[ \sqrt{x} > 5 \implies x > 25 \]
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The square root of the number must be less than 7: \[ \sqrt{x} < 7 \implies x < 49 \]
Thus, we are looking for integers \( x \) such that: \[ 25 < x < 49 \] We also need to ensure that the square root of \( x \) is irrational. Square roots are irrational for non-perfect squares. The perfect squares within our range are \( 25 \) (which we found already), \( 36 \), and \( 49 \).
The non-perfect square 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
- 48
The least integer in this list is \( 26 \).
Thus, the answer is: 26
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