1. Compute 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

 

2.  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

 

 Sutra : Yaavadunam Taavaduunikruthya vargam cha yogayet

Meaning : "Whatever the extent of its deficiency, lessen it further to that very extent; and also set up the square of that deficiency".

This sutra is a corollary of the Nikhilam sutra.

1. Consider a simple example 9²

Step 1 : Consider the nearest base (here 10).

Step 2 : As 9 has a deficiency of 1 (10 - 9 = 1), we should decrease it further by 1, and set down our LHS of the Answer as '8'.

Step 3 : On the RHS put the square of the deficiency (here 1).

 we get 9² = 81.

2. consider 102

1) Base is 100

2) Deficiency is '-2' (100 - 102 = -2)

  Therefore we subtract '-2' from 102

102 - (-2) = 104

This is our RHS

3) Our LHS now becomes (-2)² which is 4

 Since the base is 100 we write it as '04', so that we get 102² = 10404

If we have multiples or sub multiples of a base, we employ the same technique as in 'Aanurupyena'. (See Nikhilam Multiplication)

 3. Consider 28²

1) Let 20 be the Working Base and 10 as the Main Base.

 Therefore x = (Main Base)/(Working Base) = 10/20 = 1/2

2) Here the deficiency = 20 - 28 = -8

Therefore RHS = 28 - (-8) = 36

Divide by x i.e. by (1/2).

We get   36/(1/2) = 72.   This is the required RHS.

3) LHS = (-8)² = 64

Since Main Base is 10, we put only '4' on the LHS and carry over '6' to the RHS

Therefore we get

28² = 72+6 | 4 == 784

         8 is 2 below 10 and 7 is 3 below 10.

You subtract crosswise 8-3 or 7 - 2 to get 5,
the first figure of the answer.
And you multiply vertically: 2 x 3 to get 6,
the last figure of the answer

The answer is 56.

Multiply 88 by 98

Both 88 and 98 are close to 100.
88 is 12 below 100 and 98 is 2 below 100

As before the 86 comes from
subtracting crosswise: 88 - 2 = 86
(or 98 - 12 = 86: you can subtract
either way, you will always get
the same answer).
And the 24 in the answer is
just 12 x 2: you multiply vertically.
So 88 x 98 = 8624

Multiply 103 x 104 = 10712

The answer is in two parts: 107 and 12,
107 is just 103 + 4 (or 104 + 3),
and 12 is just 3 x 4.