Question

Find the square roots of 100 and 169 by the method of repeated subtraction.

Answer 1

We know that the sum of the first n odd natural numbers is n².

Consider  √100.

(i) 100 − 1 = 99 (ii) 99 − 3 = 96 (iii) 96 − 5 = 91 (iv) 91 − 7 = 84 (v) 84 − 9 = 75 (vi) 75 − 11= 64 (vii) 64 − 13 = 51 (viii) 51 − 15 = 36 (ix) 36 − 17 = 19  (x) 19 − 19 = 0

We have subtracted successive odd numbers starting from 1 to 100, and obtained 0 at 10^th step.

Therefore, √100 = 10

The square root of 169 can be obtained by the method of repeated subtraction as follows.

(i) 169 − 1 = 168 (ii) 168 − 3 = 165 (iii) 165 − 5 = 160 (iv) 160 − 7 = 153 (v) 153 − 9 = 144 (vi) 144 − 11 = 133 (vii) 133 − 13 = 120 (viii) 120 − 15 = 105 (ix) 105 − 17 = 88 (x) 88 − 19 = 69 (xi) 69 − 21 = 48 (xii) 48 − 23 = 25 (xiii)25 − 25 = 0

We have subtracted successive odd numbers starting from 1 to 169, and obtained 0 at 13^th step.

Therefore,√169 = 13