VEDIC MATHS-97

  

VEDIC MATHS

                           By OMKAR TENDOLKAR

Hello friends,

                      This is  post number 97 from the series of "Vedic maths" blogs. Here in this blog we will learn about "Raising to Fourth and Higher Powers-I"

Reference:

We had already learn about "Method Of Finding Cubes" in our previous blogs. If you have missed my last blog then please visit "VEDIC MATHS-50" to "VEDIC MATHS-52".

Raising a Number to the Fourth Power:

The method to raise a number to the fourth power is very much like the method of cubes.

So let’s see what (a + b) is when raised to the power 4.

For example, 

3^4 = 81

5^4 = 625.

(a + b)^4 = (a)^4 + 4 (a^3) b + 6 (a)^2 (b)^2 + 4 a (b^3) + (b)^4

The expression on the RHS can be broken into two parts as given below. The first part has the terms a^3, a^2b, ab^2, and b^3 and the second part has the terms 2a^2b and 2ab^2. 

   (a)^4 +   (a^3) b +   (a)^2 (b)^2  +    a (b^3) + (b)^4

+            3 (a^3) b + 5 (a)^2 (b)^2 + 3 a (b^3)    

------------------------------------------------------------------------

  (a)^4 + 4 (a^3) b + 6 (a)^2 (b)^2 + 4 a (b^3) + (b)^4

There is one interesting thing related with this formula is given below:

• If we take first term (a)^4 and multiply it with b/a gives (a^3)b i.e second term
• If we take second term (a^3)b and multiply it with b/a gives (a^2)(b^2) i.e third term
• If we take third term (a^2)(b^2) and multiply it with b/a gives a(b^3) i.e fourth term
• If we take fourth term a(b^3) and multiply it with b/a gives (b)^4 i.e fifth term

On the basis of the above observations it can be concluded that as we move from right to left the numbers are in geometric progression too and the ratio between them is a/b.

Now that we have understood this concept, let’s solve some examples. 

Example:

Say we have to find (12)^4. 

Here a is 1 and b is 2. 

Step 1:

We raise a to the power of 4, which means (1)^4 = 1. This 1 is the first digit. We put it down like this: 1.

Step 2:

We then multiply the subsequent digits by 2 as b/a is 2/1 = 2. So our first line looks like this:

1  2  4  8  16

Step 3:

We will now do the next step for the second line.

• 2 gets multiplied by 3 to become 6.
• 4 gets multiplied by 5 to become 20.
• And 8 gets multiplied by 3 to become 24.

Our sum looks like this now:

1   2    4    8   16

     6  20  24

Step 4:

In our final step, we start adding from right to left. Remember, each column will give us a single digit.

• From 16 we put down 6 in the unit’s place and carry over 1 to the next step. 
• In the ten’s place, we add 8 + 24 + 1 (carried over) = 33. We put down 3 in the ten’s place and carry over 3 to the next step.
• In the hundred’s column, we have 4 + 20 + 3, this gives us 27. We put down 7 and carry over 2 to the next step.
• In the thousand’s place, we have 2 + 6 + 2 (carried over) = 10. We put down 0 and carry 1 to the next step.
• In the final step, we have 1 + 1 = 2.
• So our final answer is 20736.

Our sum on completion looks like this:

1   2    4    8   16

     6  20  24

-----------------------

2   0    7    3     6

Answer:

(12)^4 = 20736.

Solve the following example.

1. (2)^4

Ans :  (2)^4 =16.

2. (9)^4

Ans :  (9)^4 =  6561.

3. (7)^4

Ans :  (7)^4 = 2401.

The simplicity of this method can be vouched from examples given above.

You may try following examples.

1. (5)^4

2. (8)^4

3. (3)^4

4. (4)^4

5. (8)^4

In next blog we will discuss about "Raising to Fourth and Higher Powers-II".     

Are you excited for this?...

Then, please wait for it.

I will post my new blog in next week.

We will meet very soon through our next  blog. Till that stay connected, stay healthy and stay safe.

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for giving your valuable time.

Good day😊