Vedic Math - Fourth Power of 2 Digit Numbers

Note: Vedic Math Blog has been moved to http://vedicmath.vedantatree.com/. Please bookmark the new address for new and existing blogs.

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.

If you like the article, you may contribute by:

• Posting your comments which will add value to the article contents
• Posting the article link on Social Media using the Social Media Bookmark bar
• Connecting with 'VedantaTree' on Facebook (https://www.facebook.com/VedantaTree)

Vedic Math - Cube of 2 Digit Numbers

Note: Vedic Math Blog has been moved to http://vedicmath.vedantatree.com/. Please bookmark the new address for new and existing blogs.

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
                    -------------------------
Add                     1     3     3     1
11³ = 1331