What is the square of 2352?

Answer

Verified

Hint: Prime factorize the given number and check whether each of the prime can be paired or not. If each of the prime factors can be paired then the number will be a perfect square. If not, then multiply by prime factors such that each of them can be paired.Complete step-by-step solution -

A number is called a perfect square if it can be written as the product of any number multiplied by itself.
Given number is 2352.
Prime factorization of 2352;

Step 1: Write all the prime factors together,
i.e. \[2352=2\times 2\times 2\times 2\times 3\times 7\times 7\]
Step 2: Make pairs of the same numbers.
\[\times \times \times \]
Now, we can see that each of the numbers except 3 can be paired.
This is not a perfect square because ‘3’ is left unpaired.
To make it a perfect square, we need to make a pair of this remaining 3.
So, we need to multiply the given number with 3.
After multiplying by 3, the number and prime factorization will change to,
\[\begin{align}
  & 2352\times 3=\times \times \times \\
 & \Rightarrow 7056=\times \times \times \\
\end{align}\]
Now, for getting square root of this perfect square number,
Step 3: For each pair, take one of the two in the pair to the square root
So, required square root
$\begin{align}
  & =2\times 2\times 3\times 7 \\
 & =84 \\
\end{align}$

Note: If a number has its unit digit ‘2’, it can’t be a perfect square. For a number to be a perfect square, its unit digit must be 1 or 4 or 6 or 9 or 5.

Answer

Verified

Hint: In the above question, a whole number is given. We have to find out if the number is a perfect square or not. Now, we know that this number is not a perfect square. Now, we have to find a smallest number which can be multiplied to it so that the number becomes a perfect square. Firstly, we will factorize the number given and then look for a number which can be multiplied to make this number a perfect square.

Complete step-by-step solution:
 In the above question, we are given with whole number 2352
Now, finding the square root of this number,
We get,
$ \Rightarrow \sqrt {2352} = 48.497$
Now, the square root of the number $48.497$ . So, we can say that the number is not a perfect square.
Now, we have to find a smallest number which we can multiply to the number to make it a perfect square.
Firstly, we will do the factorization of the number 2352.
Now, the factor of 2352 is $2 \times 2 \times 2 \times 2 \times 3 \times 7 \times 7$ . Now, we know that the number can only be a perfect square when the factors of the number are in pairs.
So, from the factors of the number 2352, we can see that all the factors are in pairs, except factor 3.
So, we will multiply the number 2352 with 3 to make it a perfect square.
Now, to verify that 3 is the smallest number which is multiplied to 2352 to make it a perfect square, we need to multiply 3 with the number and then find its square root.
$ \Rightarrow 3 \times 2352 = 7056$
Now, finding the square root of the number $7056$ ,
We get,
$ \Rightarrow \sqrt {7056} = 84$
Hence, the number is a perfect square and the perfect square is 84.

Note: For a number to be called as a perfect square then the square root of that number should be the whole number. If the square root is not a whole number then it will be a terminating or non terminating decimal.
There are simple tricks or ideas to find out squares and square root easily
For example - if there is any number whose unit digit or at ones place there is 5 then its square or square root will contain 5 at its ones place .
It can be observed that 225 is square root of 15
Again , we know that 25 is a square of 5.

Solution

2352=2×2×2×2×3×7×7 As the prime factor 3 has no pair, 2352 is not a perfect square. If 3 gets a pair then the number will become perfect square. So, we multiply 2352 by 3 to get, 2352×3=2×2×2×2×3×3×7×7 ∴2352×3=7056 is a perfect square. Thus, the required smallest multiple of 2352 is 7056 which is a perfect square. √7056=2×2×3×7=84

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