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6. Observe the following pattern and fill the missing numbers.
2
11 = 121
2
101 = 10201
2
10101 = 102030201
1010101 = ...................
2
.................. = 10203040504030201

7. Find the sum without adding them.
(a) 1 + 3 + 5 + 7 + 9 + 11 (b) 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21
(c) 11 + 13 + 15 + 17 + ... + 29 + 31
8. Using the given pattern, find the missing numbers.

2
4 + 5 + (20) = 21 2
2
2
2
2
2
5 + (6) + 30 = 31 2
2
(........) + 7 + 42 = (........) 2
2
2
2
2
2
(........) + 8 + 56 = (........) 2
2
2
2
8 + (........) + 72 = (........) 2
9. Express
(a) 100 as the sum of first 10 odd natural numbers
(b) 225 as the sum of first 15 odd natural numbers
10. Without actual squaring, find the value of
(a) (48) – (47) 2 (b) (25) – (24) 2 (c) (92) – (91) 2
2
2
2
2
2
(d) (142) – (141) 2 (e) (180) – (179) 2 (f) (256) – (255) 2
2
Square Root of Numbers
The square root of a number n is that number which when multiplied by itself, gives the number
n as the product. The square root of a number is denoted by the symbol (called radical sign),
2
i.e., the square root of n is written as n .
Examples: (a) 2 × 2 = 4 ⇒ Square root of 4 is 2, i.e., 4 = 2
(b) 3 × 3 = 9 ⇒ Square root of 9 is 3, i.e., 9 = 3

(c) 4 × 4 = 16 ⇒ Square root of 16 is 4, i.e., 16 = 4
(d) 5 × 5 = 25 ⇒ Square root of 25 is 5, i.e., 25 = 5

2
2
2
2
Since, (–1) = (–1) × (–1) = 1, (–2) = (–2) × (–2) = 4, (–3) = (–3) × (–3) = 9, (–4) = (–4) × (–4) = 16 etc.
Note: Thus, there are two integral square roots for each number, i.e., square roots of 1 are 1 and –1, square
roots of 4 are 2 and –2, square roots of 9 are 3 and –3, and so on. Here, we take up only positive
square root of a number.
Finding Square Roots

We know that subtraction is the inverse operation of addition and division is the inverse operation
of multiplication. Similarly, finding the square root is the inverse operation of finding square of
a number.

135 Squares and Square Roots