To find the smallest whole number by which 2268 should be multiplied to get a perfect square, we need to factorize 2268 into its prime factors. Then, we can determine which prime factors are needed to make the number a perfect square.
The prime factorization of 2268 is:
2268 = 2^2 * 3^2 * 7 * 9
To make it a perfect square, we need to include the missing factors. In this case, we are missing a factor of 2 and a factor of 3. Therefore, we need to multiply 2268 by 2 * 3 = 6.
2268 * 6 = 13608
Now, we have a perfect square number, 13608.
To find the square root of 13608, we can take the square root of each of its prime factors:
√(2^2 * 3^2 * 7 * 9)
= √2^2 * √3^2 * √7 * √9
= 2 * 3 * √7 * 3
= 6 * √7
So, the square root of the square number 13608 is 6√7.