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 the new number.

Answer 1

Step 1: Find the prime factors of the given number :

The prime factors of $2352$ are

$$\begin{gathered} 2352=2\times 2\times 2\times 2\times 3\times 7\times 7 \\ =2^2\times 2^2\times 3\times 7^2 \end{gathered}$$

Since the $3$ has no pair so $2352$ cannot be a perfect square number.

Step 2: Find the smallest number:

From the prime factor of $2352$ we can see that $3$ has no pair.

So if we multiply $2352$ with $3$ it become a perfect square number

Therefore the new number will be $2352\times 3=7056$

Step 3. Find square root of new number :

The prime factors of $7056$ are

$$\begin{gathered} 7056=2\times 2\times 2\times 2\times 3\times 3\times 7\times 7 \\ \Rightarrow 7056=2^4\times 3^2\times 7^2 \\ \Rightarrow \sqrt{7056}=\sqrt{2^4\times 3^2\times 7^2} \\ \Rightarrow \sqrt{7056}=2^2\times 3\times 7 \\ \Rightarrow \sqrt{7056}=84 \end{gathered}$$

Therefore the square root of the new number is $84$

Hence, $2352$ is not a perfect square number, so, $3$ must be multiplied to make it a perfect square no and thus the new number is $7056$. The square root of the new number is $84$.