Ex. 2. Find the square root of 53824.
Here 53824 lies between 40000 and 90000, that is between (200)^2 and (300)^2.
a b c 53824(200 + 30 + 2 = 232 40000 2a + b . . . 400 + 30 = 430 13824 12900 2(a + b)+c . . . 460+ 2 = 462 924 924
Ex. 3. Find the cube root of 614125.
Since 614125 lies between 512000 and 729000, that is between (80)^3 and (90)^3, therefore its cube root consists of two figures and lies between 80 and 90.
a + b 614125(80 + 5 = 85. 512000 3 a = 3 (80)^2 = 19200 102125 3ab = 3 80 5= 1200 b^2 = 5x5= 25 20425 102125
206. We shall now show that in extracting either the square or the cube root of any number, when a certain number of figures have been obtained by the common rule, that number may be nearly doubled by ordinary division.
207. If the square root of a number consists of 2n+1 figures, when the first n + 1 of these have been obtained by the ordinary method, the remaining n may be obtained by division.
Let x denote the given number; a the part of the square root already found, that is the first n+1 figures found by the common rule, with n ciphers annexed; x the remaining part of the root.
Then, N=a+x, N=a2 + 2ax+x^2; {N - a2}{2a}=x + {x2}{2a} (1)
