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\ 06-Jan-2025 Bharat Arora Proof-7 Reader’s Sign _______________________ Date __________

Example 25: Find the square root of 390625 using long division method.

Solution: We have
6 2 5
6 39 06 25
–36
122 306
–244
1245 6225
–6225
0

So, 390625 = 625
Example 26: Without finding the square root, find the number of digits in the square root of the
following numbers.

(a) 5184 (b) 14884 (c) 60996100
Solution: (a) 51 84 ⇒ Number of digits in square root = 2

(b) 1 48 84 ⇒ Number of digits in the square root = 3
(c) 60 99 61 00 ⇒ Number of digits in square root = 4

Example 27: Find the smallest 4-digit number which is a perfect square.
3 2
Solution: Smallest 4-digit number = 1000, 3 10 00

Since 1000 is not a perfect square, so we find a 4-digit number closest to it which is – 9
2
a perfect square. Clearly, 1000 < 32 . 62 100
– 124
Thus, 24 should be added to 1000 to obtain the smallest 4-digit perfect square. – 24
Therefore, the smallest 4-digit perfect square = 1000 + 24 = 1024.

Example 28: Find the least number which must be subtracted from 3158 to make it a perfect square.

Solution: First, we find the square root of 3158. 5 6
5 31 58
We see that the remainder is 22.
–25
\ By subtracting 22 from 3158, we get 3136 = (56) 2 106 658

Hence, the required number = 22 – 636

Example 29: Which least number must be added to 6156 to make it a perfect square? 22
Solution: 78 < 6156 and 79 = 6241 7 8
2
2
2
From the working shown alongside, we see that 6156 is greater than (78) but less 7 61 56
than (79) . –49
2
148 1256
Hence, the required least number to be added is 6241 – 6156 = 85 to make 6156 a
perfect square. –1184
72
Check: 6156 + 85 = 6241 and 6241 is Quick Check
a perfect square, 6241 = 79.
Find the greatest number of four digits which is a perfect square.

139 Squares and Square Roots