Question

"Question 6 For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square number. Also find the square root of the square number so obtained. (i) 252 (ii) 2925 (iii) 396 (iv) 2645 (v) 2800 (vi) 1620

Class 8 Squares and Square Roots Page 102"

Answer 1

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)  By prime factorization of 252, we get
252 = 2 x 2 x 3 x 3 x 7

Here, 2 and 3 are in pair, but 7 does not occur in pair.Hence, the given number is not a perfect square.

Hence, 252 needs to be divided by 7 to become a perfect square

 

252/7 =( 2 x 2) x (3 x 3) x 7/7

 

36    = (2 x 2 )x (3 x 3)

Thus, 36 has 2 pairs of equal prime factors. Hence, 36 is a perfect square & √36= 2×3= 6

Thus, the required smallest whole number by which it should be divided so as to get a perfect square number is 7 and the square root is √36= 6.

 

2) By prime factorization of 2925, we get

2925 = 5 x 5 x 3 x 3 x 13

Here, 5 and 3 are in pair, but 13 has no pair, Hence, the given number is not a perfect square.

Hence, 2925 needs to be divided by 13 to become a perfect square

 

2925/13 =( 5x 5) x (3 x 3) x 13/13

 

225= (5x 5 )x (3 x 3)

Thus, 225 has 2 pairs of equal prime factors. Hence, 225 is a perfect square & √225= 5×3= 15

 Thus, the required smallest whole number by which it should be divided so as to get a perfect square number is 13 and the square root is √225= 15.

 

 3) By prime factorization of 396, we get

396 = 2 x 2 x 3 x 3 x 11

Here, 2 and 3 are in pair, but 11 does not occur in pair. Hence, the given number is not a perfect square.

Thus, 396 needs to be divided by 11 to become a perfect square.

 

396/11 = 2 x 2 x 3 x 3 x 11/11

 

36 = (2×2)×(3×3)

Thus, 36 has 2 pairs of equal prime factors. Hence, 36 is a perfect square & √36= 2×3= 6

Thus, the required smallest whole number by which it should be divided so as to get a perfect square number is 11 and the square root is √36= 6.

 

 

4) By prime factorization of 2645, we get

2645 = 5 x 23 x 23

Here, 23 is in pair, but 5 needs a pair. Hence, the given number is not a perfect square.

 

Thus, 2645 needs to be divided by 5 to become a perfect square.

 

2645 /5= 5/5 x 23 x 23

 

529= 23× 23

Thus, 529 has 1 pairs of equal prime factors. Hence, 529 is a perfect square & √529= 23

Thus, the required smallest whole number by which it should be divided so as to get a perfect square number is 5 and the square root is √529= 23.

5) By prime factorization of 2800, we get

2800 = 2 x 2 x 7 x 10 x 10

Since, only 7 is not in pair, thus, Hence, the given number is not a perfect square.

Hence, 2800 needs to be divided by 7 to become a perfect square.

 

2800/7 = 2 x 2 x 7/7 x 10 x 10

 

400= (2×2)×(10×10)

 

Thus, 400 has 2 pairs of equal prime factors. Hence, 400 is a perfect square & √400= 2×10= 20

Thus, the required smallest whole number by which it should be divided so as to get a perfect square number is 7 and the square root is √400= 20.

 

6) By prime factorization of 1620, we get

1620 = 2 x 2 x 3 x 3 x 3 x 3 x 5

Here, 2 and 3 are in pair, but 5 is not in pair. Hence, the given number is not a perfect square.

Hence, 1620 needs to be divided by 5 to become a perfect square

 

1620 /5 =( 2 x 2) x (3 x 3) x(3 x 3) x 5/5

 

324= ( 2 x 2) x (3 x 3) x(3 x 3)

Thus, 324 has 3 pairs of equal prime factors. Hence, 324 is a perfect square & √324= 2× 3× 3= 18

Thus, the required smallest whole number by which it should be divided so as to get a perfect square number is 5 and the square root is √324= 18.

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Hope this will help you....

Answer 2

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)  By prime factorization of 252, we get

252 = 2 x 2 x 3 x 3 x 7

Here, 2 and 3 are in pair, but 7 does not occur in pair.Hence, the given number is not a perfect square.

Hence, 252 needs to be divided by 7 to become a perfect square

 

252/7 =( 2 x 2) x (3 x 3) x 7/7

 

36    = (2 x 2 )x (3 x 3)

Thus, 36 has 2 pairs of equal prime factors. Hence, 36 is a perfect square & √36= 2×3= 6

Thus, the required smallest whole number by which it should be divided so as to get a perfect square number is 7 and the square root is √36= 6.

 

2) By prime factorization of 2925, we get

2925 = 5 x 5 x 3 x 3 x 13

Here, 5 and 3 are in pair, but 13 has no pair, Hence, the given number is not a perfect square.

Hence, 2925 needs to be divided by 13 to become a perfect square

 

2925/13 =( 5x 5) x (3 x 3) x 13/13

 

225= (5x 5 )x (3 x 3)

Thus, 225 has 2 pairs of equal prime factors. Hence, 225 is a perfect square & √225= 5×3= 15

 Thus, the required smallest whole number by which it should be divided so as to get a perfect square number is 13 and the square root is √225= 15.

 

 3) By prime factorization of 396, we get

396 = 2 x 2 x 3 x 3 x 11

Here, 2 and 3 are in pair, but 11 does not occur in pair. Hence, the given number is not a perfect square.

Thus, 396 needs to be divided by 11 to become a perfect square.

 

396/11 = 2 x 2 x 3 x 3 x 11/11

 

36 = (2×2)×(3×3)

Thus, 36 has 2 pairs of equal prime factors. Hence, 36 is a perfect square & √36= 2×3= 6

Thus, the required smallest whole number by which it should be divided so as to get a perfect square number is 11 and the square root is √36= 6.

 

 

4) By prime factorization of 2645, we get

2645 = 5 x 23 x 23

Here, 23 is in pair, but 5 needs a pair. Hence, the given number is not a perfect square.

 

Thus, 2645 needs to be divided by 5 to become a perfect square.

 

2645 /5= 5/5 x 23 x 23

 

529= 23× 23

Thus, 529 has 1 pairs of equal prime factors. Hence, 529 is a perfect square & √529= 23

Thus, the required smallest whole number by which it should be divided so as to get a perfect square number is 5 and the square root is √529= 23.

5) By prime factorization of 2800, we get

2800 = 2 x 2 x 7 x 10 x 10

Since, only 7 is not in pair, thus, Hence, the given number is not a perfect square.

Hence, 2800 needs to be divided by 7 to become a perfect square.

 

2800/7 = 2 x 2 x 7/7 x 10 x 10

 

400= (2×2)×(10×10)

 

Thus, 400 has 2 pairs of equal prime factors. Hence, 400 is a perfect square & √400= 2×10= 20

Thus, the required smallest whole number by which it should be divided so as to get a perfect square number is 7 and the square root is √400= 20.

 

6) By prime factorization of 1620, we get

1620 = 2 x 2 x 3 x 3 x 3 x 3 x 5

Here, 2 and 3 are in pair, but 5 is not in pair. Hence, the given number is not a perfect square.

Hence, 1620 needs to be divided by 5 to become a perfect square

 

1620 /5 =( 2 x 2) x (3 x 3) x(3 x 3) x 5/5

 

324= ( 2 x 2) x (3 x 3) x(3 x 3)

Thus, 324 has 3 pairs of equal prime factors. Hence, 324 is a perfect square & √324= 2× 3× 3= 18

Thus, the required smallest whole number by which it should be divided so as to get a perfect square number is 5 and the square root is √324= 18.

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Hope this will help you....✍✍