Question

For each of the following, find the smallest number by which it should be multiplied so as to
get a perfect square. Also find the square root of the square number so obtained
1) 2028

please send the pic of the answer
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Answer 1

Answer:

Prime factorization method for square roots:

1.First of all find the prime factors of the given number.

2.Arrange the factor in pairs such that the two primes in each pair are equal.

3.Take one number from each pair and multiply all such numbers.

4. The product obtained in step 3 is the required square root of the given number.

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By prime factorization of 2028, we get

2028 = 2 x 2 x 3 x 13 x 13

Here, 2 and 13 are in pair, but 3 needs a pair to make 2028 a perfect square.

Thus, 2028 needs to be multiplied by 3 to become a perfect square.

2028 ×3 = (2 x 2) x (3 x 3)×(13 x 13)

 

Therefore, the number 6084 has 3 pairs of equal prime factors .

Hence, 6084 is a perfect square & √ 6084= 2×3×13=78

 

Hence, the smallest number by which 2028 must be multiplied so that the product is a perfect square is 7.

And the square root of the new number is √6084=78.

hope it's helpful☺️