To determine if 2352 is a perfect square, we need to find its square root. If the square root is a whole number, then 2352 is a perfect square. If the square root is not a whole number, then 2352 is not a perfect square.


Using a calculator, we find that the square root of 2352 is approximately 48.49. Since 48.49 is not a whole number, we can conclude that 2352 is not a perfect square.


To find the smallest multiple of 2352 which is a perfect square, we need to factor 2352 into its prime factors and then identify the missing factors to make it a perfect square.


To factor 2352, we can start by dividing it by 2 until we get an odd number. This gives us:

2352 ÷ 2 = 1176
1176 ÷ 2 = 588
588 ÷ 2 = 294
294 ÷ 2 = 147
147 ÷ 3 = 49
49 ÷ 7 = 7

Therefore, the prime factorization of 2352 is:

2352 = 2^4 × 3 × 7^2


To make 2352 a perfect square, we need to identify the missing factors. Since 2^4 is already a perfect square, we need to find the smallest values of 3 and 7 that will make the expression a perfect square.

Since the exponent of 3 is 1, we need to add another factor of 3 to make it a perfect square. Likewise, since the exponent of 7 is 2, we need to add one more factor of 7 to make it a perfect square.

Thus, the smallest multiple of 2352 which is a perfect square is:

2352 × 3 × 7 = 4704


To find the square root of 4704, we can use a calculator or factor it into its prime factors and simplify the expression.

Factorizing 4704 gives:

4704 = 2^5 × 3^2 × 7

Simplifying the expression gives:

√4704 = √(2^4 × 2 × 3^2 × 7) = 4 × 3√14

Therefore, the square root of the new number is 4 × 3√14.