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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.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo