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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)

profileAshok Gupta
2016-08-06 03:23:48 
Shortcut method to find the square of any two digit number:

suppose we want to find the square of $96$

$100-96=4$

$96-4/4×4\\
9216 $

$96^2=9216$

lets try another number

$87^2\\
100-87=13\\
87-13/13×13\\
74/169$

$74+1/69\\
7569$

Now try to square of $46$

$50-4=46$

$\dfrac{46-4}{2}/4×4\\
21/16\\
2116$
like 0 dislike 0 reply
profileRohit Sharma
2016-07-22 09:57:08 
How can we calculate square of $97$ by this method ?
like 0 dislike 0 reply
profileRaj
2016-09-02 21:06:44 
Take base $100$

$97-100=-3$
i.e., deviation is $-3$

Left side $=97+(-3)=94$

right side $=(-3)^2=9$
Since base has two zeros, write it as $09$

Hence, answer is $9409$
like 0 dislike 0
profileRajendra Krishna
2016-05-21 05:04:35 
What is the method to find out that a number is perfect square or cube ?
like 0 dislike 0 reply
profileshreyyyy
2016-01-08 14:25:12 
hey guys how to find square roots in decimal places? like $2.5, 44.5?$
any brilliant here?
like 0 dislike 0 reply
profileMrugesh
2016-08-09 18:16:49 
First of all multiply the number with $100$ and keep that in mind.

Lets take a example of $2.5$
By multiplying it with $100$ it becomes $250$

Now think of the nearest perfect square and that is in this case is $256$

Now find the difference b/w that perfect square and the given no.
the difference b/w $256$ and $250$ is $6$

Now think of the very basic expansion equation $(a-b)^2=a^2-2ab+b^2$
so now in this case, we can say that
$250=(16-x)^2=256-2×16×x+x^2$

Here $x$ is very small. So we can neglect it so.
$6=2×16×x$
So $x=\dfrac{6}{32}$ which is approx $0.1875$
So subtract it from $16$ that makes it $15.8125$

Now remind that we have multiplied the original no. with $100.$ So after rooting we have to divide the answer with $10$
so final answer is $1.58125$ approx.
hope this helps.
like 0 dislike 0
profilemonika sharma
2016-01-21 06:13:49 
Find the square of the no. simply by removing the decimals then in ans put the decimals twice before the ans.

for eg $2.5$
square of $25=625$
now in $2.5$ decimal is preceding $1$ digit
so in $625$ answer will precede $2$ digits. So final answer will be $6.25$
like 0 dislike 0
profileadi
2015-09-16 14:23:41 
$39^2=?$
Want answer with above method in steps.
If given it will be helpful for me
like 1 dislike 0 reply
profilesomesh
2015-10-27 12:30:42 
Consider the base as $40$

$39\qquad \qquad-1\\
39\qquad \qquad -1$
-----------------------
$(39-1)~/~(-1×-1)$
$~~38~\qquad /\qquad  1$   

Since it is to base $40,$ multiply only the $38$ part with $4$ $=152$
$(38×4)/1$
$152 / 1$

Ans $=1521$
like 0 dislike 0
profileRaj
2015-09-16 17:03:43 
The method explained is suitable if the number is close to a power of $10$
like 0 dislike 0
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