Cubing by Yaavadunam
This method is similar to squaring by Yaavadunam. It is just modified a bit, as we shall see in the next few examples.
Consider 133
Step 1 : Consider nearest base (here 10).
Step 2 : As 13 has a excess of '3' (13 - 10 = 3), we double the excess and add the original number (13) to it, and put it on the LHS.
Therefore we get 13 + 6 = 19
Step 3 : Now find the new excess. In this case it is 19-10 = 9. Now multiply this with the original excess to get the middle part of the answer.
Therefore we get 9 * 3 = 27
Step 4 : Now cube the original excess and put it as the last part
Carry over any big numbers and total to get the answer.
19 7 7
2 2
21 9 7
Therefore 133 = 2197
Now consider 473
As in 'Nikhilam' and Squaring, we use 'Aanurupyena' here.
1) Let the main base be 10 and the working base be 50
therefore the ratio
x = (Main Base)/(Working Base) = 10/50 = 1/5
2) Excess is -3 (47 - 50 = -3). Double the excess and add the original number (here 47) to it.
We get 47 - 6 = 41.
The Base correction for this part is achieved by dividing by x2 .
therefore we get 41/(1/25) = 41 * 25 = 1025
3) Excess in the new uncorrected number (41 - 50 = -9) is multiplied by the original excess(-3) to obtain the second part.
Therefore we get -9 * -3 = 27
The Base correction for this part is achieved by dividing by x .
therefore we get 27 * 5 = 135
4) The third part is obtained by cubing the excess.
(-3)3 = -27
5) Carry over the extra numbers and total to obtain the final answer
1025 0 0
13 5 0
-2 7
1038 2 3
Therefore the final answer is 103823