(vi) 768
reach of the following numbers, find the smallest whole number by which it should
te divided so as to get a perfect square, Also find the square root of the square
number so obtained
(iii) 396
(iv) 2645
(iv) 2925
(vi) 1620
() 2800
Answers
Step-by-step explanation:
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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Answer:
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