Example 6 Is 2352 a perfect square? If not, find the smallest multiple of 2352 which is a perfect square. Find the square root of the new number.Prime factorizing 2352
Prime factorizing 2352
Thus,
2352 = 2 × 2 × 2 × 2 × 3 × 7 × 7
Since 3 is not in pair,
Thus, we cannot calculate square root of 2352
∴ 2352 is not a perfect square
We need to find
the smallest multiple of 2352 which is a perfect square
Thus, we need to find a number multiplied to 2352 to get a perfect square
The prime factorization of 2352 is
2352 = 2 × 2 × 2 × 2 × 3 × 7 × 7
Since 3 does not occur in pair,
we multiply by 3 to make it a pair
So, our number becomes
2352 × 3 = 2 × 2 × 2 × 2 × 3 × 7 × 7 × 3
7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7
Now, it becomes a perfect square.
Hence, 7056 is the smallest multiple.
Finding the Square root of 7056
Since
7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7
Therefore,
∴ √7056 = 2 × 2 × 3 × 7
= 4 × 21
= 84
∴ The required multiple of 2352 is 7056
and its square root is 84.

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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