Solution Let us first determine the factors by the method of prime factorization So, prime factors of 176 = 2 × 2 × 2 × 2 × 11 On grouping and pairing the factors we obtain = (2 × 2) × (2 × 2) × (11) = 22 × 22 × 11 Factor 11 has no pair. ∴176 is not a perfect square. The smallest number it should be multiplied to get a perfect square is 11. Then, 176 × 11 = 1936 Factors of 1936 = 2 × 2 × 2 × 2 × 11 × 11 = √1936 = √(22 × 22 × 112) = 2 × 2 × 11 = 44 ∴Square root of 1936 is 44.
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Solution: (i) 252 = 2 x 2 x 3 x 3 x 7 Here, prime factor 7 has no pair. Therefore 252 must be multiplied by 7 to make it a perfect square. \therefore252\times7=1764 And (i) \sqrt{1764}=2\times3\times7=42  (ii) 180 = 2 x 2 x 3 x 3 x 5 Here, prime factor 5 has no pair. Therefore 180 must be multiplied by 5 to make it a perfect square. \therefore180\times5=900 And \sqrt{900}=2\times3\times5=30 (iii) 1008 = 2 x 2 x 2 x 2 x 3 x 3 x 7 Here, prime factor 7 has no pair. Therefore 1008 must be multiplied by 7 to make it a perfect square. \therefore1008\times7=7056 And \sqrt{7056}=2\times2\times3\times7=84 (iv) 2028 = 2 x 2 x 3 x 13 x 13 Here, prime factor 3 has no pair. Therefore 2028 must be multiplied by 3 to make it a perfect square. \therefore2028\times3=6084 And \sqrt{6084}=2\times2\times3\times3\times13\times13=78 (v) 1458 = 2 x 3 x 3 x 3 x 3 x 3 x 3 Here, prime factor 2 has no pair. Therefore 1458 must be multiplied by 2 to make it a perfect square. \therefore1458\times2=2916 And \sqrt{2916}=2\times3\times3\times3=54 (vi) 768 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3 Here, prime factor 3 has no pair. Tehrefore 768 must be multiplied by 3 to make it a perfect square. \therefore768\times3=2304 And \sqrt{2304}=2\times2\times2\times2\times3=48
Solution: We have to find the smallest whole number by which the number should be multiplied so as to get a perfect square number To get a perfect square, each factor of the given number must be paired. (i) 252 Hence, prime factor 7 does not have its pair. If 7 gets a pair, then the number becomes a perfect square. Therefore, 252 has to be multiplied by 7 to get a perfect square. So, perfect square is 252 × 7 = 1764 1764 = 2 × 2 × 3 × 3 × 7 × 7 Thus, √1764 = 2 × 3 × 7 = 42 (ii) 180 Hence, prime factor 5 does not have its pair. If 5 gets a pair, then the number becomes a perfect square. Therefore, 180 has to be multiplied by 5 to get a perfect square. So, perfect square is 180 × 5 = 900 900 = 2 × 2 × 3 × 3 × 5 × 5 Thus, √900 = 2 × 3 × 5 = 30 (iii) 1008 Hence, prime factor 7 does not have its pair. If 7 gets a pair, then the number becomes a perfect square. Therefore, 1008 has to be multiplied by 7 to get a perfect square. So, perfect square is 1008 × 7 = 7056 7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7 Thus, √7056 = 2 × 2 × 3 × 7 = 84 (iv) 2028 Hence, prime factor 3 does not have its pair. If 3 gets a pair, then the number becomes a perfect square. Therefore, 2028 has to be multiplied by 3 to get a perfect square. So, perfect square is 2028 × 3 = 6084 6084 = 2 × 2 × 13 × 13 × 3 × 3 Thus, √6084 = 2 × 13 × 3 = 78 (v) 1458 Hence, prime factor 2 does not have its pair. If 2 gets a pair, then the number becomes a perfect square. Therefore, 1458 has to be multiplied by 2 to get a perfect square. So, perfect square is 1458 × 2 = 2916 2916 = 3 × 3 × 3 × 3 × 3 × 3 × 2 × 2 Thus, √2916 = 3 × 3 × 3 × 2 = 54 (vi) 768 Hence, prime factor 3 does not have its pair. If 3 gets a pair, then the number becomes a perfect square. Therefore, 768 has to be multiplied by 3 to get a perfect square. So, perfect square is 768 × 3 = 2304 2304 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 Thus, √2304 = 2 × 2 × 2 × 2 × 3 = 48 ☛ Check: NCERT Solutions for Class 8 Maths Chapter 6 Video Solution: NCERT Solutions for Class 8 Maths Chapter 6 Exercise 6.3 Question 5 Summary: For each of the following numbers, (i) 252 (ii) 180 (iii) 1008 (iv) 2028 (v) 1458 (vi) 768 the smallest whole number by which it should be multiplied so as to get a perfect square number and the square root of the square number so obtained are as follows: (i) 7; √1764 = 42 (ii) 5; √900 = 30 (iii) 7; √7056 = 84 (iv) 3; √6084 = 78 (v) 2; √2916 = 54 and (vi) 3; √2304 = 48 ☛ Related Questions: |
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