To find the smallest number by which -1008 should be multiplied to get a perfect square, we need to factorize -1008.

The prime factorization of -1008 is: -1008 = -1 * 2^4 * 3^2 * 7

To make it a perfect square, we need to remove the negative sign and balance the powers of the prime factors. Since the power of 2 is 4, we need to multiply by 2^2 to make it a perfect square. Similarly, we need to multiply by 3^2 and 7 to balance the powers of 3 and 7.

So, the smallest number by which -1008 should be multiplied to get a perfect square is: 2^2 * 3^2 * 7 = 4 * 9 * 7 = 252

The square root of 252 is √252 = 15.874.

Therefore, the smallest number by which -1008 should be multiplied to get a perfect square is 252, and the square root of the square number obtained is approximately 15.874.