Question

Is 2352 a perfect square? If not, find the smallest number by

which 2352 must be multiplied so that the product is a perfect
square. Find the square root of new number.

Answer 1

The smallest number by which 2352 must be multiplied so that the product is a perfect square = 3

The square root of the new number = 84

Explanation:

√2352 ≈ 48.5 so not a perfect square

Prime factorization of 2352 yields

2352 = 2 x 2 x 2 x 2 x 7 x 7 x 3

In exponential form this is
2⁴ x 7² x 3¹

So

`√(2352) = √(2^4 \cdot 7^2 \cdot 3)\\\\= √(2^4) \cdot √(7^2) \cdot √(3)\\\\= 2^2 \cdot 7 \cdot √(3)\\\\= 28 √(3)`

To get rid of the radical in the square root and get a whole number, all you have to do is multiply
`√(2352)` by √3

`28 √(3) \cdot √(3) = 28\cdot 3 = 84\\\\84^2 = 7056 = 2352 \cdot 3\\`

This means that if you multiply 2352 by 3 it will become a perfect square

Check:

`2352 \cdot 3 = 7056\\\\√(7056) = 84`

Answer 2

No, 2352 is not a perfect square.

To find the smallest number by which 2352 must be multiplied so that the product is a perfect square, we need to factorize 2352 into its prime factors.

2352 = 2^4 x 3 x 7^2

To make it a perfect square, we need to multiply it by 2^2 and 7, which gives us:

2352 x 2^2 x 7 = 9408

Now, we can take the square root of 9408:

√9408 = √(2^8 x 3 x 7) = 2^4 x √(3 x 7) = 16√21

Therefore, the smallest number by which 2352 must be multiplied so that the product is a perfect square is 2^2 x 7, which gives us the square root of 9408 as 16√21.