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No, the square root of 24 is NOT an integer. An integer is a whole number, and the square root of 24 is 4.89897948557.
No. The square root of an integer is always either an integer or an irrational number.
No, the square root of an integer will not always be an integer. It may or may not be. The square root of nine is three, both of which are integers. The square root of two is not an integer. In fact, it is not even a rational number.
The square root of 48 is not an integer which means a whole number.
Yes and no. It depends on your definition of square root. By the actual one, yes. All non-negative numbers have a square root. That square root might be irrational but it has a square root, nonetheless. 10 isn't a square number because there's no integer that can be squared to make ten but 10 definitely has a square root: 3.16227766....... If by square root you mean an integer square root, then no. If a number has an integer as its square root then you could square that integer to get the number, making it a square number.