"Question 5 For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained. (i) 252 (ii) 180 (iii) 1008 (iv) 2028 (v) 1458 (vi) 768
Answers
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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1) We first find the prime factors of 252 By prime factorization.
Hence,
252 = 2 x 2 x 3 x 3 x 7
= (2×2)×(3×3)×7
Here, we see that 2 and 3 occur in pairs but 7 needs a pair. Hence, the given number is not a perfect square.
If We now multiply 252 by 7 then we get
252×7= 1764 = (2×2)×(3×3)×(7×7)
Therefore,the number 252 has 3 pairs of equal prime factors.
Hence, 1764 is a perfect square &
√1764= 2×3×7=42
Hence, the smallest number by which 1764 must be multiplied so that the product is a perfect square is 7.
And the square root of the new number is √1764=42.
2)
By prime factorization, we get, 180 = (3 x 3) x (2 x 2) x 5
Here, 3 and 2 are in pair but 5 needs a pair to make 180 a perfect square.
So, 180 needs to be multiplied by 5 to become a perfect square.
180×5 =(3 x 3) x (2 x 2) x (5×5)
Therefore, the number 180 has 3 pairs of equal prime factors .
Hence, 900 is a perfect square & √ 900= 3×2×5=30
Hence, the smallest number by which 180 must be multiplied so that the product is a perfect square is 5.
And the square root of the new number is √900=30.
3) By prime factorization of 1008, we get
1008 = (2 x 2) x (2 x 2) x (3 x 3 )x 7
Here, 2 and 3 are in pair, but 7 needs a pair to make 1008 a perfect square.
Thus, 1008 needs to be multiplied by 7 to become a perfect square.
1008 × 7 = (2 x 2 )x (2 x 2) x (3 x 3) x (7×7)
Therefore, the number 1008 has 4 pairs of equal prime factors .
Hence 7056 is a perfect square & √ 7056= 2×2×3×7=84
Hence, the smallest number by which1008 must be multiplied so that the product is a perfect square is 7.
And the square root of the new number is √7056=84.
4) 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.
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
Hence, the smallest number by which 1458 must be multiplied so that the product is a perfect square is 2.
And the square root of the new number is √2916=54.
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.
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Hope this will help you....