Question

Find the cube roots of the following numbers by successive subtraction of numbers : 1, 7, 19, 37, 61 , 91, 127, 169, 217, 271, 331, 397 ...
(i) 64 (ii) 512 (iii) 1728

Answer 1

$\left(i\right)6464-3=6363-7=5656-19=3737-37=064=(4{)}^3$
$\left(ii\right)512512-1=511511-7=504504-19=485485-37=448448-61=387387-91=296296-127=169169-169=0\therefore 512=(8{)}^3$
$\therefore$ Cube root of 512 = 8
$\left(iii\right)17281728-1=17271727-7=17201720-19=17011701-37=16641664-91=16031603-91=15121512-127=13851385-169=121612160217=728728-331=397397-397=0\therefore 1728=(12{)}^3$
$\therefore$ Cube root of $1728=1^2$