What is the least integer whose square root is an irrational number between 5 and 7?
Responses
25
26
6
13
The least integer whose square root is an irrational number between 5 and 7 is 26.
can you explain
Sure! 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 perfect squares nearest to 5 are 4 (2^2) and 9 (3^2), and the perfect squares nearest to 7 are 4 (2^2) and 9 (3^2).
Since we are looking for an irrational number between 5 and 7, the square root of this number cannot be an integer or any fraction made from perfect squares, like 2 or 3.
The next perfect square greater than 5 is 6^2 = 36, but its square root is 6, which is not an irrational number.
Therefore, the least integer whose square root is an irrational number between 5 and 7 is 26.
The square root of 26 is approximately 5.099, which 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 determine the smallest perfect square that is greater than 25 and less than 49. This can be done by finding the square of the smallest integer greater than 5 and the square of the largest integer less than 7.
The smallest integer greater than 5 is 6. The square of 6 is 36, which is greater than 25.
The largest integer less than 7 is 6. The square of 6 is 36, which is less than 49.
Therefore, the least integer whose square root is an irrational number between 5 and 7 is 6.
To find the least integer whose square root is an irrational number between 5 and 7, we can start by finding the square root of the upper limit (7). The square root of 7 is approximately 2.645751311. Since this is an irrational number, we know that the least integer whose square root is between 5 and 7 must be greater than 7.
Next, we find the square root of the lower limit (5). The square root of 5 is approximately 2.236067977. Since this is also an irrational number, we know that the least integer whose square root is between 5 and 7 must be greater than 5.
Since we need the least integer, we round up the square root of the lower limit to the nearest whole number, which gives us 3.
Therefore, the answer is 13 because it's the least integer greater than 7 whose square root is between 5 and 7.