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Square of a Number Using Vedic Mathematics

Let’s see how we can find square of a number faster using Vedic Mathematics (Nikhilam method)

Example 1: Find square of $12$

Step 1

$10$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=12-10=2$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=12+2=14$

Step 2

Our base $10$ has a single zero. Therefore, right side of the answer has a single digit and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=2^2=4$

Therefore, answer $=144$

Example 2: Find square of $13$

Step 1

$10$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=13-10=3$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

To find the deviation, just remove the leftmost digit

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=13+3=16$

Step 2

Our base $10$ has a single zero. Therefore, right side of the answer has a single digit and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=3^2=9$

Therefore, answer $=169$

Example 3: Find square of $14$

Step 1

$10$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=14-10=4$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

To find the deviation, just remove the leftmost digit

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=14+4=18$

Step 2

Our base $10$ has a single zero. Therefore, right side of the answer has a single digit and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=4^2=16.$ But right side of the answer can have only a single digit because our base $10$ has a only single zero. Hence, from the obtained number $16,$ we will take right side as $6$ and $1$ is taken as a carry which will be added to our left side. Hence left side becomes $18+1=19$

Therefore, answer $=196$

Example 4: Find square of $106$

Step 1

$100$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=106-100=6$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

To find the deviation, just remove the leftmost digit

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=106+6=112$

Step 2

Our base $100$ has two zeros. Therefore, right side of the answer has two digits and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=6^2=36$

Therefore, answer $=11236$

Example 5: Find square of $112$

Step 1

$100$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=112-100=12$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

To find the deviation, just remove the leftmost digit

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=112+12=124$

Step 2

Our base $100$ has two zeros. Therefore, right side of the answer has two digits and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=12^2=144.$ But right side of the answer can have only two digits because our base $100$ has only two zeros. Hence, from the obtained number $144,$ we will take right side as $44$ and $1$ is taken as a carry which will be added to our left side. Hence left side becomes $124+1=125$

Therefore, answer $=12544$

Video Tutorial

Note

This method is extremely useful for competitive examinations and if practiced well, square of a number can be determined within seconds using the same.

Comments(77)

profileAshraf
2015-06-08 01:14:17 
Using this method, how to find $78^2,82^2$ etc. Can I take $70$ and $80$ respectively for $10$ and $100$ as used in that method?
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profileangel
2015-06-04 06:44:06 
$9987^2=?$
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profileSiddharth Shankar
2015-11-20 18:03:54 
9987 = 10000-13 
so, write 9987-13 = 9974 as first four digits from the left.
now (-13)^2 = 169 ; put 0169 after 9974 (0169 because the base has four zeroes in it)

So the answer is: 99740169 !
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profileamit singh
2015-03-15 04:51:36 
Hey... 

To find square use 
(ab)^2 = a^2 / 2ab / b^ 2
For e.g -- 38^2= 3^2 / 2. 3. 8/8^2
9/48/64
Now write 4 as it is and add 6 to 48 I.e 54 
Again write 4 of 54 and add 5 to remaining 9 we get 14 so complete ans is .... 1444
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profileshivam
2015-03-12 11:47:19 
How would you find square of 55 using this method??
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profileRajesh
2016-03-13 14:58:27 
55^2= 5^2/2.5.5/5^2
25/50/25
Now write 5 as it is and add 2 to 50 i.e become 52 again 
write 2 of 52 and add 5 to remaining 25 we get 30 . 
Hence answer is : 30 25.
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profileAbcd
2015-05-03 08:57:46 
We can use another trick
55^2=first find the square in units place that is 5=25
Now we got the rhs =25
Now we have to find lhs=we have to multiply the number in units place by its ahead number that is 6=5*6=30=lhs
So 55^2=3025
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profileshivam jain
2015-04-04 06:48:51 
whenever this type of ques like 55,45,65 etc
use this
1. 55 = 5 , 5 
 in this unit digit 5 do square which is 25
and , tens digit 5 multiply to the next digit i.e 6=30
now combine the result is 3025
similarly,
45 = 4x5 , 5x5 = 20,25= 2025
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profilepg18
2015-03-12 11:36:15 
How to find out square of 99?
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profileAswin
2015-12-18 18:31:50 
100-1=99
99-1=98
Difference from 100 to 99=1
&its square 01 
i.e., ans = 9801
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