Question
What is the least integer whose square root is an irrational number between 5 and 7?
To find the least integer whose square root is an irrational number between 5 and 7, we need to find the smallest perfect square greater than 5 and the largest perfect square less than 7.
The smallest perfect square greater than 5 is 6^2 = 36, and the largest perfect square less than 7 is 2^2 = 4.
So, the least integer whose square root is an irrational number between 5 and 7 is the integer immediately greater than 4, which is 5.
Therefore, the answer is 5.
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To find the least integer whose square root is an irrational number between 5 and 7, we need to determine the perfect squares that fall within this range and then find the smallest integer whose square root is not a rational number.
The perfect squares between 5 and 7 are 4 (2^2) and 9 (3^2).
Taking the square root of 4, we get √4 = 2, which is a rational number. Taking the square root of 9, we get √9 = 3, which is also a rational number.
The next perfect square after 9 is 16 (4^2). Taking the square root of 16, we get √16 = 4, which is a rational number.
Since all the perfect squares between 5 and 7 have rational square roots, there is no integer in this range whose square root is an irrational number. Therefore, there is no least integer that satisfies this condition.
To find the least integer whose square root is an irrational number between 5 and 7, we can start by finding the square roots of the numbers 5 and 7.
The square root of 5 is approximately 2.236 and the square root of 7 is approximately 2.646. Since we're looking for an irrational number between these two values, we need to find the smallest perfect square greater than 5 and the largest perfect square smaller than 7.
The smallest perfect square greater than 5 is 9 (3^2), and the largest perfect square smaller than 7 is 4 (2^2). So, we're looking for the least integer whose square root is greater than the square root of 4 (2) and less than the square root of 9 (3).
Starting from 2 and increasing the integer values, we find that the least integer whose square root falls within this range is 3. The square root of 3 is approximately 1.732.
Therefore, the least integer whose square root is an irrational number between 5 and 7 is 3.