RS Aggarwal Class 8 Math Third Chapter Squares and Square Roots Exercise 3A Solution
EXERCISE 3A
(1) Using the prime factorization method, find which of the following numbers are perfect squares:
(i) 441 = 3 × 3 × 7 × 7 = 32 × 72
(ii) 576 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 = 26 × 32
(iii) 11025 = 5 × 5 × 3 × 3 × 7 × 7 = 52 × 32 × 72
(iv) 1176 = 2 × 2 × 2 × 3 × 7 × 7
(v) 5625 = 3 × 3 × 5 × 5 × 5 × 5 = 32 × 54
(vi) 9075 = 3 × 5 × 5 × 11 × 11
(vii) 4225 = 5 × 5 × 13 × 13 = 52 × 132
(viii) 1089 = 3 × 3 × 11 × 11 = 32 × 112
(2) Show that each of the following numbers is a perfect square. In each case, find the number whose square is the given number:
(i) 1225 = 5 × 5 × 7 × 7 = 52 × 72
Thus, 1225 is the product of pairs of equal factors.
∴ 1225 is a perfect square.
Also = (5 × 7)2 = (35)2
Hence, 35 is the number whose square is 1225.
(ii) 2601 = 3 × 3 × 17 × 17 = 32 × 172
Thus, 2601 is the product of pairs of equal factors.
∴ 2601 is a perfect square.
Also = (3 × 17)2 = (51)2
Hence, 51 is the number whose square is 2601.
(iii) 5929 = 7 × 7 × 11 × 11 = 72 × 112
Thus, 5929 is the product of pairs of equal factors.
∴ 5929 is a perfect square.
Also = (7 × 11)2 = (77)2
Hence, 77 is the number whose square is 5929.
(iv) 7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7 = 22 × 22 × 32 × 72
Thus, 7056 is the product of pairs of equal factors.
∴ 7056 is a perfect square.
Also = (2 × 2 × 3 × 7)2 = (84)2
Hence, 84 is the number whose square is 7056.
(v) 8281 = 7 × 7 × 13 × 13 = 72 × 132
Thus, 8281 is the product of pairs of equal factors.
∴ 8281 is a perfect square.
Also = (7 × 13)2 = (91)2
Hence, 91 is the number whose square is 8281.
(3) By what least number should the given number be multiplied to get a perfect square number? In each case, find the number whose square is the new number.
(i) 3675
Solution: Resolving 3675 into prime factors, we get
3675 = 3 × 5 × 5 × 7 × 7 = (3 × 52 × 72)
Thus, to get a perfect square number, the given number should be multiplied by 3.
New number = (32 × 52 × 72) = (3 × 5 × 7)2 = (105)2
Hence, the number whose square is the new number = 105.
(ii) 2156
Solution: Resolving 2156 into prime factors, we get
2156 = 2 × 2 × 7 × 7 × 11
Thus, to get a perfect square number, the given number should be multiplied by 11.
New number = (22 × 72 × 112) = (2 × 7 × 11)2 = (154)2
Hence, the number whose square is the new number = 154.
(iii) 3332
Solution: Resolving 3332 into prime factors, we get
3332 = 2 × 2 × 7 × 7 × 17
Thus, to get a perfect square number, the given number should be multiplied by 17.
New number = (22 × 72 × 172) = (2 × 7 × 17)2 = (238)2
Hence, the number whose square is the new number = 238.
(iv) 2925
Solution: Resolving 2925 into prime factors, we get
2925 = 3 × 3 × 5 × 5 × 13
Thus, to get a perfect square number, the given number should be multiplied by 13.
New number = (32 × 52 × 132) = (3 × 5 × 13)2 = (195)2
Hence, the number whose square is the new number = 195.
(v) 9075
Solution: Resolving 9075 into prime factors, we get
9075 = 3 × 5 × 5 × 11 × 11
Thus, to get a perfect square number, the given number should be multiplied by 3.
New number = (32 × 52 × 112) = (3 × 5 × 11)2 = (165)2
Hence, the number whose square is the new number = 165.
(vi) 7623
Solution: Resolving 7623 into prime factors, we get
7623 = 3 × 3 × 7 × 11 × 11
Thus, to get a perfect square number, the given number should be multiplied by 7.
New number = (32 × 72 × 112) = (3 × 7 × 11)2 = (231)2
Hence, the number whose square is the new number = 231.
(vii) 3380
Solution: Resolving 3380 into prime factors, we get
3380 = 2 × 2 × 5 × 13 × 13
Thus, to get a perfect square number, the given number should be multiplied by 5.
New number = (22 × 52 × 132) = (2 × 5 × 13)2 = (130)2
Hence, the number whose square is the new number = 130.
(viii) 2475
Solution: Resolving 2475 into prime factors, we get
2475 = 3 × 3 × 5 × 5 × 11
Thus, to get a perfect square number, the given number should be multiplied by 11.
New number = (32 × 52 × 112) = (3 × 5 × 11)2 = (165)2
Hence, the number whose square is the new number = 165.
(4) By what least number should the given number be divided to get a perfect square number? In each case, find the number whose square is the new number.
(i) 1575
Solution: Resolving 1575 into prime factors, we get
1575 = 3 × 3 × 5 × 5 × 7 = (32 × 52 × 7)
Thus, to get a perfect square number, the given number should be divided by 7.
New number obtained = (32 × 52) = (3 × 5)2 = (15)2
Hence, the number whose square is the new number = 15.
(ii) 9075
Solution: Resolving 9075 into prime factors, we get
9075 = 3 × 5 × 5 × 11 × 11 = (3 × 52 × 112)
Thus, to get a perfect square number, the given number should be divided by 3.
New number obtained = (52 × 112) = (5 × 11)2 = (55)2
Hence, the number whose square is the new number = 55.
(iii) 4851
Solution: Resolving 4851 into prime factors, we get
4851 = 3 × 3 × 7 × 7 × 11 = (32 × 72 × 11)
Thus, to get a perfect square number, the given number should be divided by 11.
New number obtained = (32 × 72) = (3 × 7)2 = (21)2
Hence, the number whose square is the new number = 21.
(iv) 3380
Solution: Resolving 3380 into prime factors, we get
3380 = 2 × 2 × 5 × 13 × 13 = (22 × 5 × 132)
Thus, to get a perfect square number, the given number should be divided by 5.
New number obtained = (22 × 132) = (2 × 13)2 = (26)2
Hence, the number whose square is the new number = 26.
(v) 4500
Solution: Resolving 4500 into prime factors, we get
4500 = 2 × 2 × 3 × 3 × 5 × 5 × 5 = (22 × 32 × 5 × 52)
Thus, to get a perfect square number, the given number should be divided by 5.
New number obtained = (22 × 32 × 52) = (2 × 3 × 5)2 = (30)2
Hence, the number whose square is the new number = 30.
(vi) 7776
Solution: Resolving 7776 into prime factors, we get
7776 = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 = (22 × 22 × 2 × 3 × 32 × 32)
Thus, to get a perfect square number, the given number should be divided by 2 × 3.
New number obtained = (22 × 22 × 32 × 32) = (2 × 2 × 3 × 3)2 = (36)2
Hence, the number whose square is the new number = 36.
(vii) 8820
Solution: Resolving 8820 into prime factors, we get
8820 = 2 × 2 × 3 × 3 × 5 × 7 × 7 = (22 × 32 × 5 × 72)
Thus, to get a perfect square number, the given number should be divided by 5.
New number obtained = (22 × 32 × 72) = (2 × 3 × 7)2 = (42)2
Hence, the number whose square is the new number = 42.
(viii) 4056
Solution: Resolving 4056 into prime factors, we get
4500 = 2 × 2 × 2 × 3 × 13 × 13 = (22 × 2 × 3 × 132)
Thus, to get a perfect square number, the given number should be divided by 2 × 3.
New number obtained = (22 × 132) = (2 × 13)2 = (26)2
Hence, the number whose square is the new number = 26.
(5) Find the largest number of 2 digits which is a perfect square.
Ans: The largest 2 digits number is 99.
Square of 10 = 100 > 99, thus the number would be less than 10.
And the largest whole number less than 10 is 9.
Therefore, 9 × 9 = 81
(6) Find the largest number of 3 digits which is a perfect square.
Ans: The largest three digits number is 999. But 961 is a largest three digits number, is a perfect square.
961 = 31 × 31
Here, for easy to understand we take the before and after number of 31. Those are 30 and 32 respectively.
Now, 30 × 30 = 900 and 32 × 32 = 1024.
Hence, we can write 961 is largest three numbers has a perfect square.
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