### Vedic Math - Fourth Power of 2 Digit Numbers

We discussed the cube of 2-digit number in previous article. In this article, we shall describe the fourth power of 2-digit numbers using the same formula.

The Algebraic Expression of (a + b)4

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3+ b4

We can rewrite the above equation as:
a4       a3b          a2b2           ab3        b4
3a3b        5a2b2          3ab3
So, apply the same rule which we applied in previous article, while finding cubic of the number. Consider the first term as a4 and the remaining terms get multiplied by b/a with the previous term.
The Difference comes in second row, in fourth power, we multiply 2nd and 4th term by 3 and 3rd term by 5.

Example: 114

1    1    1    1    1
3    5    3
-------------------------
1    4    6    4    1
-------------------------

Example: 324

81     54      36     24     16
162    180     72
-------------------------------------
104      8       5        7       6
-------------------------------------

The "Binomial Theorem" is thus capable of practical application more comprehensively in Vedic Math. Here it is been utilised for splendid purpose as described above, with Vedic Sutras.

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### Vedic Math - Cube of 2 Digit Numbers

Cube of numbers plays an important role in mathematics calculations, like in finding cube root of the numbers. So it is useful if we can do the cube of numbers quickly. We are trying for the same here using 'Anurupya Sutra' of Vedic Math. Let us learn how we can do it.

Cubes of the single digits i.e. from (1 to 9) are given below:
13  = 1,              23  = 8,              33  = 27,            43  = 64,            53  = 125,
63  = 216,          73  = 343,          83  = 512,           93  = 729,          103  = 1000

If we observe closely, the last digit of every cubic number is unique i.e. numbers from (1 to 9) does not repeat. This observation will be very helpful while calculating cube roots of any 2-digit numbers.

Let us first see the Algebraic Expression for Cube root:

(a + b)3 = a3 + 3a2b + 3ab2+ b3

Above expression for cube root of (a + b) contain 4 terms in total.
• 1st term is  a3
• 2nd term is a2b =  a3  x (b/a)  = 1st term x (b/a)
• 3rd term is ab2 = a2b x (b/a) = 2nd term x (b/a)
• 4th term is b3   = ab2 x (b/a) = 3rd term x (b/a)

Here (b/a) is the common ratio

Also, as the whole, 2nd term is 3a2b = a2b + 2a2b          {split as sum of two terms}
and, 3rd term is 3ab2 = ab2 + 2ab2           {split as sum of two terms}

So to find the cube, we have to compute a3 and b/a.

In Vedic Math, same formula can be used in a different way to find the cube of 2-digit numbers i.e. ab. Apply formula on 'ab' like (a+b)3 as stated above, and add the results of different rows in vertical columns. You will be able to do the cube of any two digit numbers quickly.

We shall use 'Anurupya Sutra' of Vedic Math for this cube calculation, which states:
"If you start with the cube of first digit and take the next three numbers (in the top row) in a Geometrical Proportion (in the ratio of original digits themselves), you will find that the fourth figure on the right hand will be just the cube of second digit".

Following is the step by step description of finding the cube of 2-digit number:
• Step 1: In the first row, start with a^3 as 1st term and multiplying 1st term by (b/a) to get 2nd term.
• Step 2: Repeat the multiplication till 4th term.
• Step 3: In the second row, double the two middle terms (i.e. 2nd term and 3rd term) and write just below 2nd term and 3rd term.
• Step 4: Add them vertically in columns. Carry forward the 10th place digit to next column.

The example given below will describe this method well.

Example: 113
Here a = 1 , b = 1 ,  a3 = 1 ,  b/a = 1/1 = 1          (Here common ratio is equal to 1)

Now see the formation of the table:
First Row             1     1     1     1
Second Row               2     2
-------------------------
113 = 1331

Example: 133
Here a = 1 , b = 3 ,  a3 = 1 ,  b/a = 3/1 = 3          (Here common ratio is greater than 1)

Now see the formation of the table:
First Row              1     3     9     27       (Note: 4th term is just  b3  as shown above in algebraic
Second Row                6     18               expression)
-------------------------
-------------------------
=   1     9      7     7       (Apply carry over rule)
2     2
=   1      9      9     7
2
=   1      1     9     7
1
=   2      1     9     7
133= 2197

Example: 523
Here a = 5 , b = 2 ,  a3 = 125 ,  b/a = 2/5            (Here common ratio is less than 1)

Now see the formation of the table:
First Row                 125     50     20     8
Second Row                     100     40
---------------------------
---------------------------
=  125       0        0     8        (Apply carry over rule)
15        6
=  140      6    0    8
523= 140608

Some more examples are as follows:
(1) 163 =  1    6    36    216
12    72
-----------------------
4    0     9      6
-----------------------

(2) 323 =  27    18    12     8
36    24
-----------------------
32     7      6      8
-----------------------

(3) 973 =  729    567      441      343
1113      882
-----------------------------
912       6          7          3
-----------------------------
or better way for number near base, 973 = (100-3)3    where a=100, b= -3 and b/a= -3/100
= 1000000    -30000      900    -27
=                   -60000    1800
------------------------------------
1000000    -90000    2700    -27
------------------------------------
= 912673

Hope it will help!!
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