Question

how to find the smallest number by which we multiply a number to make it a perfect square no.
like for 1458 and 768

Answer 1

Answer:

BY PRIME FACTORIZATION

Step-by-step explanation:

.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.

5) By prime factorization of 1458, we get

1458 = 2 x 3 x 3 x 3 x 3 x 3 x 3

Here, 3 are in pair, but 2 needs a pair to make 1458 a perfect square.

So, 1458 needs to be multiplied by 2 to become a perfect square.

1458 ×2 =(2× 2) x (3 x 3) x (3 x 3) x (3 x 3)

 

Therefore, the number  has 4 pairs of equal prime factors .

Hence, 2916 is a perfect square & √2916= 2×3×3×3=54

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6) By prime factorization of 768, we get

768= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3

Here, 2 are in pair, but 3 needs a pair to make 768 a perfect square.

So, 768 needs to be multiplied by 3 to become a perfect square.

768 × 3=( 2 x 2 )x (2 x 2) x (2 x 2) x (2 x 2) x (3×3)

 

Therefore, the number 768 has 5 pairs of equal prime factors .

Hence, 2304 is a perfect square & √2304= 2×2×2×2×3=48

 

Hence, the smallest number by which 768 must be multiplied so that the product is a perfect square is 3

And the square root of the new number is √2304=48.

HOPE THIS WILL SURELY HELP YOU